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LECTURES   ON   MATHEMATICS 


1 


THE  EYANSTON  COLLOQUIUM 


LECTURES  ON  MATHEMATICS 

DELIVERED 

FROM  AUG.  28  TO  SEPT.  9,  1893 

BEFORE  MEMBERS  OF  THE  CONCHESS  OF  MA  THEM  A  TICS 

HELD  IN  CONNECTION  WITH  THE   WORLD'S 

FAIR  IN  CHICAGO 

AT    NORTHWESTERN    UNIVERSITY 
EVANSTON,  ILL. 

BY 

FELIX    KLEIN 


REPORTED  BY  ALEXANDER  ZIWET 


REPUBLISHED   BY   THE 

AMERICAN    MATHEMATICAL    SOCIETY 

NEW  YORK 
1911 

All  rights  reserved 


COPYRIGHT,  1893, 
BY  MACMILLAN  AND  CO. 


NortoooU 

J.  S.  Gushing  &  Co.  — Berwick  &  Smith. 
Boston,  Mass.,  U.S.A. 


Sciences 
Library 


1 


To  reproduce  after  a  lapse  of  seventeen  years  lectures  which 
at  the  time  they  were  delivered  were  in  such  close  contact  with 
the  most  recent  work  of  that  day,  may  well  call  for  a  word  of 
justification.  Has  mathematics  not  advanced  since  then,  and 
are  the  questions  here  treated  of  first  importance  at  the  present 
time  ?  I  reply  by  asking :  What  is  important  in  the  develop- 
ment of  mathematics  ?  Is  it  solely  the  attainment  of  new  re- 
sults of  potential  value,  or  must  not  an  essential  part  of  the  best 
scientific  efforts  of  each  new  age  be  devoted  to  possessing  itself 
of  the  heritage  of  the  age  that  has  just  preceded  it  ? 

It  is  a  noble  example  that  Professor  Klein  has  set  all  ages  of 
mathematicians  that,  beginning  his  activity  at  a  time  when  the 
contributions  of  the  immediate  past  were  so  rich  and  so  unre- 
lated, he  was  able  to  uncover  the  essential  bonds  that  connect 
them  and  to  discern  the  fields  to  whose  development  the  new 
methods  were  best  adapted. 

His  instinct  for  that  which  is  vital  in  mathematics  is  sure, 
and  the  light  with  which  his  treatment  illumines  the  problems 
here  considered  may  well  serve  as  a  guide  for  the  youth  who  is 
approaching  the  study  of  the  problems  of  a  later  day. 

WILLIAM   F.   OSGOOD. 
CAMBRIDGE,  December  31,  1910. 


918576 


PREFACE. 

THE  Congress  of  Mathematics  held  under  the  auspices  of 
the  World's  Fair  Auxiliary  in  Chicago,  from  the  2ist  to  the 
26th  of  August,  1893,  was  attended  by  Professor  Felix  Klein 
of  the  University  of  Gottingen,  as  one  of  the  commissioners  of 
the  German  university  exhibit  at  the  Columbian  Exposition. 
After  the  adjournment  of  the  Congress,  Professor  Klein  kindly 
consented  to  hold  a  colloquium  on  mathematics  with  such  mem- 
bers of  the  Congress  as  might  wish  to  participate.  The  North- 
western University  at  Evanston,  111.,  tendered  the  use  of  rooms 
for  this  purpose  and  placed  a  collection  of  mathematical  books 
from  its  library  at  the  disposal  of  the  members  of  the  collo- 
quium. The  following  is  a  list  of  the  members  attending  the 
colloquium  :  — 

W.  W.  BEMAN,  A.M.,  professor  of  mathematics,  University  of  Michigan. 

E.  M.  BLAKE,  Ph.D.,  instructor  in  mathematics,  Columbia  College. 

O.  BOLZA,  Ph.D.,  associate  professor  of  mathematics,  University  of  Chicago. 
H.  T.  EDDY,  Ph.D.,  president  of  the  Rose  Polytechnic  Institute. 
A.  M.  ELY,  A.B.,  professor  of  mathematics,  Vassar  College. 

F.  FRANKLIN,  Ph.D.,  professor  of  mathematics,  Johns  Hopkins  University. 
T.  F.  HOLGATE,  Ph.D.,  instructor  in  mathematics,  Northwestern  University. 
L.  S.  HULBURT,  A.M.,  instructor  in  mathematics,  Johns  Hopkins  University. 
F.  H.  LOUD,  A.B.,  professor  of  mathematics  and  astronomy,  Colorado  College. 
J.  McMAHON,  A.M.,  assistant  professor  of  mathematics,  Cornell  University. 
H.   MASCHKE,   Ph.D.,   assistant   professor   of   mathematics,   University  of 

Chicago. 
E.  H.  MOORE,  Ph.D.,  professor  of  mathematics,  University  of  Chicago. 


viii  PREFACE. 

J.  E.  OLIVER,  A.M.,  professor  of  mathematics,  Cornell  University. 

A.  M.  SAWIN,  Sc.M.,  Evanston. 

W.  E.  STORY,  Ph.D.,  professor  of  mathematics,  Clark  University. 

E.  STUDY,  Ph.D.,  professor  of  mathematics,  University  of  Marburg. 

H.  TABER,  Ph.D.,  assistant  professor  of  mathematics,  Clark  University. 

H.  W.  TYLER,  Ph.D.,  professor  of  mathematics.  Massachusetts  Institute  of 
Technology. 

J.  M.  VAN  VLECK,  A.M.,  LL.D.,  professor  of  mathematics  and  astronomy, 
Wesleyan  University. 

E.  B.  VAN  VLECK,  Ph.D.,  instructor  in  mathematics,  University  of  Wis- 
consin. 

C.  A.  WALDO,  A.M.,  professor  of  mathematics,  De  Pauw  University. 

H.  S.  WHITE,  Ph.D.,  associate  professor  of  mathematics,  Northwestern  Uni- 
versity. 

M.  F.  WINSTON,  A.B.,  honorary  fellow  in  mathematics,  University  of  Chicago. 

A.  ZIWET,  assistant  professor  of  mathematics,  University  of  Michigan. 

The  meetings  lasted  from  August  28th  till  September  9th  ; 
and  in  the  course  of  these  two  weeks  Professor  Klein  gave  a 
daily  lecture,  besides  devoting  a  large  portion  of  his  time  to 
personal  intercourse  and  conferences  with  those  attending  the 
meetings.  The  lectures  were  delivered  freely,  in  the  English 
language,  substantially  in  the  form  in  which  they  are  here 
given  to  the  public.  The  only  change  made  consists  in  oblit- 
erating the  conversational  form  of  the  frequent  questions  and 
discussions  by  means  of  which  Professor  Klein  understands  so 
well  to  enliven  his  discourse.  My  notes,  after  being  written 
out  each  day,  were  carefully  revised  by  Professor  Klein  him- 
self, both  in  manuscript  and  in  the  proofs. 

As  an  appendix  it  has  been  thought  proper  to  give  a  transla- 
tion of  the  interesting  historical  sketch  contributed  by  Professor 
Klein  to  the  work  Die  deutschen  Universitaten.  The  translation 
was  prepared  by  Professor  H.  W.  Tyler,  of  the  Massachusetts 
Institute  of  Technology. 

It  is  to  be  hoped  that  the  proceedings  of  the  Chicago  Con- 
gress of  Mathematics,  in  which  Professor  Klein  took  a  leading 


PREFACE.  ix 

part,  will  soon  be  published  in  full.  The  papers  presented  to 
this  Congress,  and  the  discussions  that  followed  their  reading, 
form  an  important  complement  to  the  Evanston  colloquium. 
Indeed,  in  reading  the  lectures  here  published,  it  should  be  kept 
in  mind  that  they  followed  immediately  upon  the  adjournment 
of  the  Chicago  meeting,  and  were  addressed  to  members  of  the 
Congress.  This  circumstance,  in  addition  to  the  limited  time 
and  the  informal  character  of  the  colloquium,  must  account 
for  the  incompleteness  with  which  the  various  subjects  are 
treated. 

In  concluding,  the  editor  wishes  to  express  his  thanks  to 
Professors  W.  W.  Beman  and  H.  S.  White  for  aid  in  preparing 
the  manuscript  and  correcting  the  proofs. 

ALEXANDER  ZIWET. 
ANN  ARBOR,  MICH.,  November,  1893. 

The  new  edition  is  printed  from  the  original  plates.     The 
only  changes  in  the  text  consist  in  the  correction  of  a  few 

misprints. 

A.  Z. 
ANN  ARBOR,  January,  1911. 


CONTENTS. 

LECTURE  PAGE 

I.     Clebsch i 

II.     Sophus  Lie 9 

III.  Sophus  Lie        ..........  18 

IV.  On  the  Real  Shape  of  Algebraic  Curves  and  Surfaces          .         .  25 
V.     Theory  of  Functions  and  Geometry    .         ...         .         .         -33 

VI.     On  the   Mathematical    Character   of  Space-Intuition,  and   the 

Relation  of  Pure  Mathematics  to  the  Applied  Sciences    .         .  41 

VII.     The  Transcendency  of  the  Numbers  e  and  TT      ....  51 

VIII.     Ideal  Numbers 58 

IX.     The  Solution  of  Higher  Algebraic  Equations      ....  67 

X.     On  Some  Recent  Advances  in  Hyperelliptic  and  Abelian  Func- 
tions         75 

XI.     The  Most  Recent  Researches  in  Non-Euclidean  Geometry         .  85 

XII.     The  Study  of  Mathematics  at  Gottingen 94 

The  Development  of  Mathematics  at  the  German  Universities   .  99 


LECTURES   ON   MATHEMATICS. 


LECTURE  I. :   CLEBSCH. 

(August  28,  1893.) 

IT  will  be  the  object  of  our  Colloquia  to  pass  in  review  some 
of  the  principal  phases  of  the  most  recent  development  of  math- 
ematical thought  in  Germany. 

A  brief  sketch  of  the  growth  of  mathematics  in  the  German 
universities  in  the  course  of  the  present  century  has  been  con- 
tributed by  me  to  the  work  Die  deutschen  Universitdten,  com- 
piled and  edited  by  Professor  Lexis  (Berlin,  Asher,  1893),  for 
the  exhibit  of  the  German  universities  at  the  World's  Fair.* 
The  strictly  objective  point  of  view  that  had  to  be  adopted  for 
this  sketch  made  it  necessary  to  break  off  the  account  about 
the  year  1870.  In  the  present  more  informal  lectures  these 
restrictions  both  as  to  time  and  point  of  view  are  abandoned. 
It  is  just  the  period  since  1870  that  I  intend  to  deal  with,  and 
I  shall  speak  of  it  in  a  more  subjective  manner,  insisting  par- 
ticularly on  those  features  of  the  development  of  mathematics 
in  which  I  have  taken  part  myself  either  by  personal  work  or 
by  direct  observation. 

The  first  week  will  be  devoted  largely  to  Geometry,  taking 
this  term  in  its  broadest  sense ;  and  in  this  first  lecture  it  will 
surely  be  appropriate  to  select  the  celebrated  geometer  Clebsch 

*  A  translation  of  this  sketch  will  be  found  in  the  Appendix,  p.  99. 


2  LECTURE   1. 

as  the  central  figure,  partly  because  he  was  one  of  my  principal 
teachers,  and  also  for  the  reason  that  his  work  is  so  well  known 
in  this  country. 

Among  mathematicians  in  general,  three  main  categories  may 
be  distinguished ;  and  perhaps  the  names  logicians,  formalists, 
and  intuitionists  may  serve  to  characterize  them,  (i)  The  word 
logician  is  here  used,  of  course,  without  reference  to  the  mathe- 
matical logic  of  Boole,  Peirce,  etc. ;  it  is  only  intended  to  indi- 
cate that  the  main  strength  of  the  men  belonging  to  this  class 
lies  in  their  logical  and  critical  power,  in  their  ability  to  give 
strict  definitions,  and  to  derive  rigid  deductions  therefrom. 
The  great .  and  wholesome  influence  exerted  in  Germany  by 
Weierstrass  in  this  direction  is  well  known.  (2)  The  formalists 
among  the  mathematicians  excel  mainly  in  the  skilful  formal 
treatment  of  a  given  question,  in  devising  for  it  an  "algorithm." 
Gordan,  or  let  us  say  Cayley  and  Sylvester,  must  be  ranged  in 
this  group.  (3)  To  the  intuitionists,  finally,  belong  those  who 
lay  particular  stress  on  geometrical  intuition  (Anschauung),  not 
ir;  pure  geometry  only,  but  in  all  branches  of  mathematics. 
What  Benjamin  Peirce  has  called  "  geometrizing  a  mathematical 
question  "  seems  to  express  the  same  idea.  Lord  Kelvin  and 
von  Staudt  may  be  mentioned  as  types  of  this  category. 

Clebsch  must  be  said  to  belong  both  to  the  second  and  third 
of  these  categories,  while  I  should  class  myself  with  the  third, 
and  also  the  first.  For  this  reason  my  account  of  Clebsch's 
work  will  be  incomplete ;  but  this  will  hardly  prove  a  serious 
drawback,  considering  that  the  part  of  his  work  characterized 
by  the  second  of  the  above  categories  is  already  so  fully  appre- 
ciated here  in  America.  In  general,  it  is  my  intention  here, 
not  so  much  to  give  a  complete  account  of  any  subject,  as  to 
supplement  the  mathematical  views  that  I  find  prevalent  in  this 
country. 


CLEBSCH.  3 

As  the  first  achievement  of  Clebsch  we  must  set  down  the 
introduction  into  Germany  of  the  work  done  previously  by 
Cayley  and  Sylvester  in  England.  But  he  not  only  trans- 
planted to  German  soil  their  theory  of  invariants  and  the  inter- 
pretation of  projective  geometry  by  means  of  this  theory;  he 
also  brought  this  theory  into  live  and  fruitful  correlation"  with 
the  fundamental  ideas  of  Riemann's  theory  of  functions.  In 
the  former  respect,  it  may  be  sufficient  to  refer  to  Clebsch's 
Vorlesungen  iiber  Geometric,  edited  and  continued  by  Linde- 
mann ;  to  his  Bindre  algebraische  Formen,  and  in  general  to 
what  he  did  in  co-operation  with  Gordan.  A  good  historical 
account  of  his  work  will  be  found  in  the  biography  of  Clebsch 
published  in  the  Math.  Annalen,  Vol.  7. 

Riemann's  celebrated  memoir  of  1857*  presented  the  new 
ideas  on  the  theory  of  functions  in  a  somewhat  startling  novel 
form  that  prevented  their  immediate  acceptance  and  recogni- 
tion. He  based  the  theory  of  the  Abelian  integrals  and  their 
inverse,  the  Abelian  functions,  on  the  idea  of  the  surface  now 
so  well  known  by  his  name,  and  on  the  corresponding  funda- 
mental theorems  of  existence  (Existenztheoreme).  Clebsch,  by 
taking  as  his  starting-point  an  algebraic  curve  defined  by  its 
equation,  made  the  theory  more  accessible  to  the  mathema- 
ticians of  his  time,  and  added  a  more  concrete  interest  to  it 
by  the  geometrical  theorems  that  he  deduced  from  the  theory 
of  Abelian  functions.  Clebsch's  paper,  Ueber  die  Anwendung 
der  AbeV  scJten  Functionen  in  der  Geometrie,^  and  the  work  of 
Clebsch  and  Gordan  on  Abelian  functions,^  are  well  known  to 
American  mathematicians ;  and  in  accordance  with  my  plan,  I 
proceed  to  give  merely  some  critical  remarks. 

*  Theorie  der  AbeFschen  Functionen,  Journal  fur  reine  und  angewandte  Mathe- 
matik,  Vol.  54  (1857),  pp.  115-155;  reprinted  in  Riemann's  Werke,  1876,  pp.  81-135. 
t  Journal  fur  reine  und  angewandte  Mathematik,  Vol.  63  (1864),  pp.  189-24.3. 
\  Theori'  der  AbePschen  Functionen,  Leipzig,  Teubner,  rR66. 


4  LECTURE   I. 

However  great  the  achievement  of  Clebsch's  in  making 
the  work  of  Riemann  more  easy  of  access  to  his  contempo- 
raries, it  is  my  opinion  that  at  the  present  time  the  book  of 
Clebsch  is  no  longer  to  be  considered  as  the  standard  work 
for  an  introduction  to  the  study  of  Abelian  functions.  The 
chief  objections  to  Clebsch's  presentation  are  twofold  :  they 
can  be  briefly  characterized  as  a  lack  of  mathematical  rigour 
on  the  one  hand,  and  a  loss  of  intuitiveness,  of  geometrical 
perspicuity,  on  the  other.  A  few  examples  will  explain  my 
meaning. 

(a)  Clebsch  bases  his  whole  investigation  on  the  considera- 
tion of  what  he  takes  to  be  the  most  general  type  of  an 
algebraic  curve,  and  this  general  curve  he  assumes  as  having 
only  double  points,  but  no  other  singularities.  To  obtain  a 
sure  foundation  for  the  theory,  it  must  be  proved  that  any 
algebraic  curve  can  be  transformed  rationally  into  a  curve 
having  only  double  points.  This  proof  was  not  given  by 
Clebsch  ;  it  has  since  been  supplied  by  his  pupils  and  follow- 
ers, but  the  demonstration  is  long  and  involved.  See  the 
papers  by  Brill  and  Nother  in  the  Math.  Annalen,  Vol.  7 
(1874),*  and  by  Nother,  ib.,  Vol.  23  (1884)-! 

Another  defect  of  the  same  kind  occurs  in  connection  with 
the  determinant  of  the  periods  of  the  Abelian  integrals.  This 
determinant  never  vanishes  as  long  as  the  curve  is  irredu- 
cible. But  Clebsch  and  Gordan  neglect  to  prove  this ;  and 
however  simple  the  proof  may  be,  this  must  be  regarded  as 
an  inexactness. 

The  apparent  lack  of  critical  spirit  which  we  find  in  the  work 
of  Clebsch  is  characteristic  of  the  geometrical  epoch  in  which 

*  Ueber  die  algebraischen  Functional  und  ihre  Anivendung  in  der  Geometric^ 
pp.  269-310. 

t  Rationale  Aus/uhrung  der  Operationen  in  der  Theorie  der  algebraischen  Func- 
tionen,  pp.  311-358. 


CLEBSCH. 


5 


he  lived,  the  epoch  of  Steiner,  among  others.  It  detracts  in  no- 
wise from  the  merit  of  his  work.  But  the  influence  of  the 
theory  of  functions  has  taught  the  present  generation  to  be 
more  exacting. 

(b)  The  second  objection  to  adopting  Clebsch's  presentation 
lies  in  the  fact  that,  from  Riemann's  point  of  view,  many  points 
of  the  theory  become  far  more  simple  and  almost  self-evident, 
whereas  in  Clebsch's  theory  they  are  not  brought  out  in  all 
their  beauty.  An  example  of  this  is  presented  by  the  idea  of 
the  deficiency  p.  In  Riemann's  theory,  where  p  represents  the 
order  of  connectivity  of  the  surface,  the  invariability  of  p  under 
any  rational  transformation  is  self-evident,  while  from  the  point 
of  view  of  Clebsch  this  invariability  must  be  proved  by  means 
of  a  long  elimination,  without  affording  the  true  geometrical 
insight  into  its  meaning. 

For  these  reasons  it  seems  to  me  best  to  begin  the  theory 
of  Abelian  functions  with  Riemann's  ideas,  without,  however, 
neglecting  to  give  later  the  purely  algebraical  developments. 
This  method  is  adopted  in  my  paper  on  Abelian  functions ;  * 
it  is  also  followed  in  the  work  Die  elliptischen  Modulfunctionen, 
Vols.  I.  and  II.,  edited  by  Dr.  Fricke.  A  general  account  of  the 
historical  development  of  the  theory  of  algebraic  curves  in  con- 
nection with  Riemann's  ideas  will  be  found  in  my  (lithographed) 
lectures  on  Riemannsche  Flachen,  delivered  in  1891-92.! 

If  this  arrangement  be  adopted,  it  is  interesting  to  follow 
out  the  true  relation  that  the  algebraical  developments  bear 
to  Riemann's  theory.  Thus  in  Brill  and  Nother's  theory,  the 
so-called  fundamental  theorem  of  Nother  is  of  primary  impor- 


*  Zur  Theorie  der  AbeVschen  Functionen,  Math.  Annalen,  Vol.  36  (1890),  pp. 
l-83. 

t  My  lithographed  lectures  frequently  give  only  an  outline  of  the  subject,  omit- 
ting details  and  long  demonstrations,  which  are  supposed  to  be  supplied  by  the 
student  by  private  reading  and  a  study  of  the  literature  of  the  subject. 


6  LECTURE   I. 

tance.  It  gives  a  rule  for  deciding  under  what  conditions  an 
algebraic  rational  integral  function  f  of  x  and  y  can  be  put  into 
the  form 


where  <f>  and  i|r  are  likewise  rational  algebraic  functions.  Each 
point  of  intersection  of  the  curves  <£  =  o  and  i/r=o  must  of 
course  be  a  point  of  the  curve  f=o.  But  there  remains  the 
question  of  multiple  and  singular  points  ;  and  this  is  disposed 
of  by  Nother's  theorem.  Now  it  is  of  great  interest  to  in- 
vestigate how  these  relations  present  themselves  when  the 
starting-point  is  taken  from  Riemann's  ideas. 

One  of  the  best  illustrations  of  the  utility  of  adopting 
Riemann's  principles  is  presented  by  the  very  remarkable 
advance  made  recently  by  Hurwitz,  in  the  theory  of  algebraic 
curves,  in  particular  his  extension  of  the  theory  of  algebraic 
correspondences,  an  account  of  which  is  given  in  the  second 
volume  of  the  Elliptische  Modulfunctionen.  Cayley  had  found 
as  a  fundamental  theorem  in  this  theory  a  rule  for  determining 
the  number  of  self-corresponding  points  for  algebraic  corre- 
spondences of  a  simple  kind.  A  whole  series  of  very  valuable 
papers  by  Brill,  published  in  the  Math.  Annalen,*  is  devoted 
to  the  further  investigation  and  demonstration  of  this  theorem. 
Now  Hurwitz,  attacking  the  problem  from  the  point  of  view 
of  Riemann's  ideas,  arrives  not  only  at  a  more  simple  and 
quite  general  demonstration  of  Cayley's  rule,  but  proceeds  to  a 
complete  study  of  all  possible  algebraic  correspondences.  He 
finds  that  while  for  general  curves  the  correspondences  consid- 

*  Ueber  z-wei  Beruhrungsprobleme,  Vol.  4  (1871),  pp.  527-549.  —  Ueber  Ent- 
sprechen  von  Punktsystemen  auf  einer  Curve,  Vol.  6  (1873),  pp.  33-65.  —  Ueber  die 
Correspondenzformel,  Vol.  7  (1874),  pp.  607-622.  —  Ueber  algebraische  Correspon- 
denten,  Vol.  31  (1888),  pp.  374-409.  —  Ueber  algebraische  Correspondenzen.  Zweite 
Abhandlung:  Specialgruppen  von  Punkten  einer  algebraischen  Curve,  Vol.  36  (1890), 
pp.  321-360. 


CLEBSCH.  7 

ered  by  Cayley  and  Brill  are  the  only  ones  that  exist,  in  the 
case  of  singular  curves  there  are  other  correspondences  which 
also  can  be  treated  completely.  These  singular  curves  are 
characterized  by  certain  linear  relations  with  integral  coeffi- 
cients, connecting  the  periods  of  their  Abelian  integrals. 

Let  us  now  turn  to  that  side  of  Clebsch's  method  which 
appears  to  me  to  be  the  most  important,  and  which  certainly 
must  be  recognized  as  being  of  great  and  permanent  value ; 
I  mean  the  generalization,  obtained  by  Clebsch,  of  the  whole 
theory  of  Abelian  integrals  to  a  theory  of  algebraic  functions 
with  several  variables.  By  applying  the  methods  he  had 
developed  for  functions  of  the  form  f(x,  y)  —  o,  or  in  homo- 
geneous co-ordinates,  f(x^  x.^  x^)  =  o,  to  functions  with  four 
homogeneous  variables  f(x^,  x^,  xs,  x^)—o,  he  found  in  1868, 
that  there  also  exists  a  number  /  that  remains  invariant  under 
all  rational  transformations  of  the  surface  f  —  o.  Clebsch 
arrives  at  this  result  by  considering  double  integrals  belonging 
to  the  surface. 

It  is  evident  that  this  theory  could  not  have  been  found  from 
Riemann's  point  of  view.  There  is  no  difficulty  in  conceiving  a 
four-dimensional  Riemann  space  corresponding  to  an  equation 
f(x,  y,  z)=o.  But  the  difficulty  would  lie  in  proving  the 
"theorems  of  existence"  for  such  a  space;  and  it  may  even  be 
doubted  whether  analogous  theorems  hold  in  such  a  space. 

While  to  Clebsch  is  due  the  fundamental  idea  of  this 
grand  generalization,  the  working  out  of  this  theory  was 
left  to  his  pupils  and  followers.  The  work  was  mainly  carried 
on  by  Nother,  who  showed,  in  the  case  of  algebraic  surfaces, 
the  existence  of  more  than  one  invariant  number  /  and  of 
corresponding  moduli,  i.e.  constants  not  changed  by  one-to-one 
transformations.  Italian  and  French  mathematicians,  in  partic- 
ular Picard  and  Poincar6,  have  also  contributed  largely  to  the 
further  development  of  the  theory. 


8  LECTURE   I. 

If  the  value  of  a  man  of  science  is  to  be  gauged  not  by  his 
general  activity  in  all  directions,  but  solely  by  the  fruitful  new 
ideas  that  he  has  first  introduced  into  his  science,  then  the 
theory  just  considered  must  be  regarded  as  the  most  valuable 
work  of  Clebsch. 

In  close  connection  with  the  preceding  are  the  general  ideas 
put  forth  by  Clebsch  in  his  last  memoir,*  ideas  to  which  he 
himself  attached  great  importance.  This  memoir  implies  an 
application,  as  it  were,  of  the  theory  of  Abelian  functions  to 
the  theory  of  differential  equations.  It  is  well  known  that  the 
central  problem  of  the  whole  of  modern  mathematics  is  the 
study  of  the  transcendental  functions  defined  by  differential 
equations.  Now  Clebsch,  led  by  the  analogy  of  his  theory  of 
Abelian  integrals,  proceeds  somewhat  as  follows.  Let  us  con- 
sider, for  example,  an  ordinary  differential  equation  of  the  first 
order  f(x,  y,  y')=o,  where  f  represents  an  algebraic  function. 
Regarding  y1  as  a  third  variable  z,  we  have  the  equation  of  an 
algebraic  surface.  Just  as  the  Abelian  integrals  can  be  classi- 
fied according  to  the  properties  of  the  fundamental  curve  that 
remain  unchanged  under  a  rational  transformation,  so  Clebsch 
proposes  to  classify  the  transcendental  functions  defined  by 
the  differential  equations  according  to  the  invariant  properties 
of  the  corresponding  surf  aces  f=o  under  rational  one-to-one 
transformations. 

The  theory  of  differential  equations  is  just  now  being  culti- 
vated very  extensively  by  French  mathematicians  ;  and  some 
of  them  proceed  precisely  from  this  point  of  view  first  adopted 
by  Clebsch. 


*  Ueber  tin  neues  Grundgebilde  der  analytischen  Gtometrie  der  Ebene,  Math. 
Annalen,  Vol.  6  (1873),  pp.  203-215. 


LECTURE  II.:   SOPHUS   LIE. 

(August  29,  1893.) 

To  fully  understand  the  mathematical  genius  of  Sophus  Lie, 
one  must  not  turn  to  the  books  recently  published  by  him  in 
collaboration  with  Dr.  Engel,  but  to  his  earlier  memoirs,  written 
during  the  first  years  of  his  scientific  career.  There  Lie  shows 
himself  the  true  geometer  that  he  is,  while  in  his  later  publi- 
cations, finding  that  he  was  but  imperfectly  understood  by  the 
mathematicians  accustomed  to  the  analytical  point  of  view,  he 
adopted  a  very  general  analytical  form  of  treatment  that  is  not 
always  easy  to  follow. 

Fortunately,  I  had  the  advantage  of  becoming  intimately 
acquainted  with  Lie's  ideas  at  a  very  early  period,  when  they 
were  still,  as  the  chemists  say,  in  the  "nascent  state,"  and 
thus  most  effective  in  producing  a  strong  reaction.  My  lecture 
to-day  will  therefore  be  devoted  chiefly  to  his  paper,  "  Ueber 
Complexe,  insbesondere  Linien-  und  Kugel-Complexe,  mil  Anw en- 
dung  auf  die  Theorie  partieller  Differentialgleichungen."  * 

To  define  the  place  of  this  paper  in  the  historical  develop- 
ment of  geometry,  a  word  must  be  said  of  two  eminent  geome- 
ters of  an  earlier  period:  Pliicker  (1801-68)  and  Monge  (1746- 
1818).  Plucker's  name  is  familiar  to  every  mathematician, 
through  his  formulae  relating  to  algebraic  curves.  But  what  is 
of  importance  in  the  present  connection  is  his  generalized  idea 

*  Math.  Annalen,  Vol.  5  (1872),  pp.  145-256. 
9 


I0  LECTURE   II. 

of  the  space-element.  The  ordinary  geometry  with  the  point  as 
element  deals  with  space  as  three-dimensioned,  conformably  to 
the  three  constants  determining  the  position  of  a  point.  A  dual 
transformation  gives  the  plane  as  element ;  space  in  this  case 
has  also  three  dimensions,  as  there  are  three  independent  con- 
stants in  the  equation  of  the  plane.  If,  however,  the  straight 
line  be  selected  as  space-element,  space  must  be  considered  as 
four-dimensional,  since  four  independent  constants  determine 
a  straight  line.  Again,  if  a  quadric  surface  F%  be  taken  as 
element,  space  will  have  nine  dimensions,  because  every  such 
element  requires  nine  quantities  for  its  determination,  viz.  the 
nine  independent  constants  of  the  surface  F2 ;  in  other  words, 
space  contains  <x> 9  quadric  surfaces.  This  conception  of  hyper- 
spaces  must  be  clearly  distinguished  from  that  of  Grassmann 
and  others.  Pliicker,  indeed,  rejected  any  other  idea  of  a  space 
of  more  than  three  dimensions  as  too  abstruse. — The  work 
of  Monge  that  is  here  of  importance,  is  his  Application  de 
r analyse  a  la  gtomttrie,  1809  (reprinted  1850),  in  which  he 
treats  of  ordinary  and  partial  differential  equations  of  the  first 
and  second  order,  and  applies  these  to  geometrical  questions 
such  as  the  curvature  of  surfaces,  their  lines  of  curvature, 
geodesic  lines,  etc.  The  treatment  of  geometrical  problems  by 
means  of  the  differential  and  integral  calculus  is  one  feature  of 
this  work ;  the  other,  perhaps  even  more  important,  is  the  con- 
verse of  this,  viz.  the  application  of  geometrical  intuition  to 
questions  of  analysis. 

Now  this  last  feature  is  one  of  the  most  prominent  character- 
istics of  Lie's  work;  he  increases  its  power  by  adopting  Pliicker's 
idea  of  a  generalized  space-element  and  extending  this  funda- 
mental conception.  A  few  examples  will  best  serve  to  give  an 
idea  of  the  character  of  his  work  ;  as  such  an  example  I  select 
(as  I  have  done  elsewhere  before)  Lie's  sphere-geometry  (Kugel- 
geometrie). 


SOPHUS  •  LIE.  1  1 

Taking  the  equation  of  a  sphere  in  the  form 

x2  -\-y*  -f-  zr  —  2  Bx  —  2  Cy  —  2  Dz  -f  E  =  o, 

the  coefficients,  j?,  C,  D,  E,  can  be  regarded  as  the  co-ordinates 
of  the  sphere,  and  ordinary  space  appears  accordingly  as  a 
manifoldness  of  four  dimensions.  For  the  radius,  R,  of  the 
sphere  we  have 


as  a  relation  connecting  the  fifth  quantity,  R,  with  the  four  co- 
ordinates, B,  C,  D,  E. 

To  introduce  homogeneous  co-ordinates,  put 

B=*,    C=c-,    D=d-,    E=e-,   R=r-; 
a  a  a  a  a 

then  a  :  b  :c  :  d:  e  are  the  five  homogeneous  co-ordinates  of  the 
sphere,  and  the  sixth  quantity  r  is  related  to  them  by  means  of 
the  homogeneous  equation  of  the  second  degree, 

?3  =  &*  +  <*+<i*-ae.  (i) 

Sphere-geometry  has  been  treated  in  two  ways  that  must  be 
carefully  distinguished.  In  one  method,  which  we  may  call  the 
elementary  sphere-geometry,  only  the  five  co-ordinates  a:b:c:d\e 
are  used,  while  in  the  other,  the  higher,  or  Lie  's,  sphere-geometry, 
the  quantity  r  is  introduced.  In  this  latter  system,  a  sphere 
has  six  homogeneous  co-ordinates,  a,  b,  c,  d,  e,  r,  connected  by 
the  equation  (i). 

From  a  higher  point  of  view  the  distinction  between  these 
two  sphere-geometries,  as  well  as  their  individual  character,  is 
best  brought  out  by  considering  the  group  belonging  to  each. 
Indeed,  every  system  of  geometry  is  characterized  by  its  group, 
in  the  meaning  explained  in  my  Erlangen  Programm  ;  *  i.e. 

*  Vergleichende  Betrachtungen  iiber  neuere  geometrische  Forschungen.  Programm 
zum  Eintritt  in  die  philosophische  Facultat  und  den  Senat  der  K.  Friedrich-Alexan- 


12  LECTURE   II. 

every  system  of  geometry  deals  only  with  such  relations  of 
space  as  remain  unchanged  by  the  transformations  of  its  group. 
In  the  elementary  sphere-geometry  the  group  is  formed  by 
all  the  linear  substitutions  of  the  five  quantities  a,  b,  c,  d,  e, 
that  leave  unchanged  the  homogeneous  equation  of  the  second 

degree 

J»  +  <«  +  ,/»  _  a*  =  o.  (2) 

This  gives  tx>  K~K  =  oo 10  substitutions.  By  adopting  this  defi- 
nition we  obtain  point-transformations  of  a  simple  character. 
The  geometrical  meaning  of  equation  (2)  is  that  the  radius  is 
zero.  Every  sphere  of  vanishing  radius,  i.e.  every  point,  is 
therefore  transformed  into  a  point.  Moreover,  as  the  polar 

2bl>'  +  2cc'  +  2dd'  —  ae'— a'e  =  o 

remains  likewise  unchanged  in  the  transformation,  it  follows 
that  orthogonal  spheres  are  transformed  into  orthogonal  spheres. 
Thus  the  group  of  the  elementary  sphere-geometry  is  character- 
ized as  the  conformal  group,  well  known  as  that  of  the  trans- 
formation by  inversion  (or  reciprocal  radii)  and  through  its 
applications  in  mathematical  physics. 

Darboux    has    further    developed    this    elementary    sphere- 
geometry.     Any  equation  of  the  second  degree 

F(a,  b,  c,  d,  e)  =  o, 

taken  in  connection  with  the  relation  (2)  represents  a  point- 
surface  which  Darboux  has  called  cyclide.  From  the  point  of 
view  of  ordinary  projective  geometry,  the  cyclide  is  a  surface  of 
the  fourth  order  containing  the  imaginary  circle  common  to  all 
spheres  of  space  as  a  double  curve.  A  careful  investigation 

ders-Universitat  zu  Erlangen.  Erlangen,  Deichert,  1872.  For  an  English  transla- 
tion, by  Haskell,  see  the  Bulletin  of  the  New  York  Mathematical  Society,  Vol.  2 
(1893),  PP-  215-249. 


SOPHUS    LIE.  I3 

of  these  cyclides  will  be  found  in  Darboux's  Lemons  sur  la 
thtorie  ge"nerale  des  surfaces  et  les  applications  ge'ome'triqiies  du 
calcnl  infinitesimal,  and  elsewhere.  As  the  ordinary  surfaces  of 
the  second  degree  can  be  regarded  as  special  cases  of  cyclides, 
we  have  here  a  method  for  generalizing  the  known  properties 
of  quadric  surfaces  by  extending  them  to  cyclides.  Thus  Mr. 
M.  Bocher,  of  Harvard  University,  in  his  dissertation,*  has 
treated  the  extension  of  a  problem  in  the  theory  of  the  poten- 
tial from  the  known  case  of  a  body  bounded  by  surfaces  of 
the  second  degree  to  a  body  bounded  by  cyclides.  A  more 
extended  publication  on  this  subject  by  Mr.  Bocher  will  appear 
in  a  few  months  (Leipzig,  Teubner). 

In  the  higher  sphere-geometry  of  Lie,  the  six  homogeneous 
co-ordinates  a:b:c:d:e:r  are  connected,  as  mentioned  above, 
by  the  homogeneous  equation  of  the  second  degree, 


The  corresponding  group  is  selected  as  the  group  of  the 
linear  substitutions  transforming  this  equation  into  itself.  We 
have  thus  a  group  of  oo36"2^^15  substitutions.  But  this  is 
not  a  group  of  point-transformations ;  for  a  sphere  of  radius 
zero  becomes  a  sphere  whose  radius  is  in  general  different  from 
zero.  Thus,  putting  for  instance 

B'  =  £,  C'  =  C,  D'  =  D,  E'=E,  R'  =  R  +  const., 

it  appears  that  the  transformation  consists  in  a  mere  dilatation 
or  expansion  of  each  sphere,  a  point  becoming  a  sphere  of 
given  radius. 

The  meaning  of  the  polar  equation 

z  bb'  +  2  cc]  +  2  dd1  —  2  rr'  —  ae'  —  a'e  =  o 

*  Ueber  die  Reihenenhuickelungen  der  Potentialtheorie,  gekronte  Preisschrift, 
Gottingen,  Dieterich,  1891. 


I4  LECTURE   II. 

remaining  invariant  for  any  transformation  of  the  group,  is  evi- 
dently that  the  spheres  originally  in  contact  remain  in  contact. 
The  group  belongs  therefore  to  the  important  class  of  contact- 
transformations,  which  will  soon  be  considered  more  in  detail. 

In  studying  any  particular  geometry,  such  as  Lie's  sphere- 
geometry,  two  methods  present  themselves. 

(i)  We  may  consider  equations  of  various  degrees  and  inquire 
what  they  represent.  In  devising  names  for  the  different  con- 
figurations so  obtained,  Lie  used  the  names  introduced  by 
Pliicker  in  his  line-geometry.  Thus  a  single  equation, 

F(a,  b,  c,  d,  e,  r)  =  o, 

is  said  to  represent  a  complex  of  the  first,  second,  etc.,  degree, 
according  to  the  degree  of  the  equation  ;  a  complex  contains, 
therefore,  oo3  spheres.  Two  such  equations, 


represent  a  congruency  containing  oo  2  spheres.    Three  equations, 
F!  =  o,     F2  =  o,     F3  =  o, 

may  be  said  to  represent  a  set  of  spheres,  the  number  being  oo1. 
It  is  to  be  noticed  that  in  each  case  the  equation  of  the  second 

degree, 

p  +  c2  +  </2  -  r*  -  ae  =  o, 

is  understood  to  be  combined  with  the  equation  F  =  o. 

It  may  be  well  to  mention  expressly  that  the  same  names  are 
used  by  other  authors  in  the  elementary  sphere-geometry,  where 
their  meaning  is,  of  course,  different. 

(2)  The  other  method  of  studying  a  new  geometry  consists 
in  inquiring  how  the  ordinary  configurations  of  point-geometry 
can  be  treated  by  means  of  the  new  system.  This  line  of 
inquiry  has  led  Lie  to  highly  interesting  results. 


SOPHUS   LIE.  !j 

In  ordinary  geometry  a  surface  is  conceived  as  a  locus  of 
points ;  in  Lie's  geometry  it  appears  as  the  totality  of  all  the 
spheres  having  contact  with  the  surface.  This  gives  a  threefold 
infinity  of  spheres,  or  a  complex  of  spheres, 

f(a,  b,  c,  d,  e,  r)  =  o. 

But  this,  of  course,  is  not  a  general  complex  ;  for  not  every  com- 
plex will  be  such  as  to  touch  a  surface.  It  has  been  shown 
that  the  condition  that  must  be  fulfilled  by  a  complex  of 
spheres,  if  all  its  spheres  are  to  touch  a  surface,  is  the  following  : 

\db)  +  \dc)  +  \dd)  ~\dr)  ~Hal)e  = 

To  give  at  least  one  illustration  of  the  further  development  of 
this  interesting  theory,  I  will  mention  that  among  the  infinite 
number  of  spheres  touching  the  surface  at  any  point  there  are 
two  having  stationary  contact  with  the  surface ;  they  are  called 
the  principal  spheres.  The  lines  of  curvature  of  the  surface 
can  then  be  defined  as  curves  along  which  the  principal  spheres 
touch  the  surface  in  two  successive  points. 

Pliicker's  line-geometry  can  be  studied  by  the  same  two 
methods  just  mentioned.  In  this  geometry  let  /12,  />13,  />14,  /34, 
/42»  /23  b£  the  usual  six  homogeneous  co-ordinates,  where 
/«=  —pK.  Then  we  have  the  identity 

/12/34  +/13/42  4-/14/23  =  °> 

and  we  take  as  group  the  QO  15  linear  substitutions  transforming 
this  equation  into  itself.  This  group  corresponds  to  the  totality 
of  collineations  and  reciprocations,  i.e.  to  the  projective  group. 
The  reason  for  this  lies  in  the  fact  that  the  polar  equation 


expresses  the  intersection  of  the  two  lines  /,  p'-. 


T6  LECTURE   II. 

Now  Lie  has  instituted  a  comparison  of  the  highest  interest 
between  the  line-geometry  of  Pliicker  and  his  own  sphere- 
geometry.  In  each  of  these  geometries  there  occur  six  homo- 
geneous co-ordinates  connected  by  a  homogeneous  equation  of 
the  second  degree.  The  discriminant  of  each  equation  is  differ- 
ent from  zero.  It  follows  that  we  can  pass  from  either  of  these 
geometries  to  the  other  by  linear  substitutions.  Thus,  to  trans- 
form 

PvtPu  +  /13/42 

into 


it  is  sufficient  to  assume,  say, 

pu  =  b  +  iV,      pK  =  d+  r,      pu  =  -a, 


It  follows  from  the  linear  character  of  the  substitutions  that 
the  polar  equations  are  likewise  transformed  into  each  other. 
Thus  we  have  the  remarkable  result  that  two  spheres  that  touch 
correspond  to  two  lines  that  intersect. 

It  is  worthy  of  notice  that  the  equations  of  transformation 
involve  the  imaginary  unit  /  ;  and  the  law  of  inertia  of  quadratic 
forms  shows  at  once  that  this  introduction  of  the  imaginary 
cannot  be  avoided,  but  is  essential. 

To  illustrate  the  value  of  this  transformation  of  line-geometry 
into  sphere-geometry,  and  vice  versa,  let  us  consider  three 
linear  equations, 

^l  =  o,  ^,  =  0,  ^3  =  0, 

the  variables  being  either  line  co-ordinates  or  sphere  co-ordi- 
nates. In  the  former  case  the  three  equations  represent  a  set 
of  lines  ;  i.e.  one  of  the  two  sets  of  straight  lines  of  a  hyper- 
boloid  of  one  sheet.  It  is  well  known  that  each  line  of  either 
set  intersects  all  the  lines  of  the  other.  Transforming  to  sphere- 


SOPHUS    LIE.  i7 

geometry,  we  obtain  a  set  of  spheres  corresponding  to  each 
set  of  lines ;  and  every  sphere  of  either  set  must  touch  every 
sphere  of  the  other  set.  This  gives  a  configuration  well 
known  in  geometry  from  other  investigations ;  viz.  all  these 
spheres  envelop  a  surface  known  as  Dupin's  cyclide.  We 
have  thus  found  a  noteworthy  correlation  between  the  hyper- 
boloid  of  one  sheet  and  Dupin's  cyclide. 

Perhaps  the  most  striking  example  of  the  fruitfulness  of  this 
work  of  Lie's  is  his  discovery  that  by  means  of  this  transfor- 
mation the  lines  of  curvature  of  a  surface  are  transformed  into 
asymptotic  lines  of  the  transformed  surface,  and  vice  versa. 
This  appears  by  taking  the  definition  given  above  for  the  lines 
of  curvature  and  translating  it  word  for  word  into  the  language 
of  line-geometry.  Two  problems  in  the  infinitesimal  geome- 
try of  surfaces,  that  had  long  been  regarded  as  entirely  distinct, 
are  thus  shown  to  be  really  identical.  This  must  certainly  be 
regarded  as  one  of  the  most  elegant  contributions  to  differential 
geometry  made  in  recent  times. 


LECTURE  III.:   SOPHUS   LIE. 

(August  30,  1893.) 

THE  distinction  between  analytic  and  algebraic  functions, 
so  important  in  pure  analysis,  enters  also  into  the  treatment 
of  geometry. 

Analytic  functions  are  those  that  can  be  represented  by  a 
power  series,  convergent  within  a  certain  region  bounded  by 
the  so-called  circle  of  convergence.  Outside  of  this  region 
the  analytic  function  is  not  regarded  as  given  a  priori ;  its 
continuation  into  wider  regions  remains  a  matter  of  special 
investigation  and  may  give  very  different  results,  according  to 
the  particular  case  considered. 

On  the  other  hand,  an  algebraic  function,  w  =  Alg.  (z),  is 
supposed  to  be  known  for  the  whole  complex  plane,  having  a 
finite  number  of  values  for  every  value  of  z. 

Similarly,  in  geometry,  we  may  confine  our  attention  to  a 
limited  portion  of  an  analytic  curve  or  surface,  as,  for  instance, 
in  constructing  the  tangent,  investigating  the  curvature,  etc. ; 
or  we  may  have  to  consider  the  whole  extent  of  algebraic  curves 
and  surfaces  in  space. 

Almost  the  whole  of  the  applications  of  the  differential  and 
integral  calculus  to  geometry  belongs  to  the  former  branch  of 
geometry  ;  and  as  this  is  what  we  are  mainly  concerned  with  in 
the  present  lecture,  we  need  not  restrict  ourselves  to  algebraic 
functions,  but  may  use  the  more  general  analytic  functions 
confining  ourselves  always  to  limited  portions  of  space.  I 

18 


SOPHUS    LIE.  lg 

thought  it  advisable  to  state  this  here  once  for  all,  since  here  in 
America  the  consideration  of  algebraic  curves  has  perhaps  been 
too  predominant. 

The  possibility  of  introducing  new  elements  of  space  has  been 
pointed  out  in  the  preceding  lecture.  To-day  we  shall  use  again 
a  new  space-element,  consisting  of  an  infinitesimal  portion  of  a 
surface  (or  rather  of  its  tangent  plane)  with  a  definite  point  in 
it.  This  is  called,  though  not  very  properly,  a  surface-element 
(Flachenelement),  and  may  perhaps  be  likened  to  an  infinitesi- 
mal fish-scale.  From  a  more  abstract  point  of  view  it  may  be 
defined  as  simply  the  combination  of  a  plane  with  a  point  in  it. 

As  the  equation  of  a  plane  passing  through  a  point  (x,  y,  z) 
can  be  written  in  the  form 


x  \y  'y  z1  being  the  current  co-ordinates,  we  have  x,  y,  z,  p,  q  as  the 
co-ordinates  of  our  surface-element,  so  that  space  becomes  a 
fivefold  manifoldness.  If  homogeneous  co-ordinates  be  used, 
the  point  (*j,  xv  x^  x^  and  the  plane  (ttlt  «2>  M&  u^i  passing 
through  it  are  connected  by  the  condition 


=  O 


expressing  their  united  position  ;  and  the  number  of  indepen- 
dent constants  is  3  +  3—1=5,  as  before. 

Let  us  now  see  how  ordinary  geometry  appears  in  this 
representation.  A  point,  being  the  locus  of  all  surface-elements 
passing  through  it,  is  represented  as  a  manifoldness  of  two 
dimensions,  let  us  say  for  shortness,  an  M2.  A  curve  is  repre- 
sented by  the  totality  of  all  those  surface-elements  that  have 
their  point  on  the  curve  and  their  plane  passing  through  the 
tangent  ;  these  elements  form  again  an  M2.  Finally,  a  surface 
is  given  by  those  surface-elements  that  have  their  point  on  the 


20  LECTURE    III. 

surface  and  their  plane  coincident  with  the  tangent  plane  of  the 
surface ;  they,  too,  form  an  Mz. 

Moreover,  all  these  Mz's  have  an  important  property  in 
common  :  any  two  consecutive  surface-elements  belonging  to 
the  same  point,  curve,  or  surface  always  satisfy  the  condition 

dz  —pdx  —  qdy  =  o, 

which  is  a  simple  case  of  a  Pfaffian  relation  ;  and  conversely,  if 
two  surface-elements  satisfy  this  condition,  they  belong  to  the 
same  point,  curve,  or  surface,  as  the  case  may  be. 

Thus  we  have  the  highly  interesting  result  that  in  the  geome- 
try of  surface-elements  points  as  well  as  curves  and  surfaces  are 
brought  under  one  head,  being  all  represented  by  twofold  mani- 
foldnesses  having  the  property  just  explained.  This  definition 
is  the  more  important  as  there  are  no  other  Mz's  having  the 
same  property. 

We  now  proceed  to  consider  the  very  general  kind  of  trans- 
formations called  by  Lie  contact-transformations.  They  are 
transformations  that  change  our  element  (x,  y,  z,  p,  q)  into 
(x',y't  z',p',  q'}  by  such  substitutions 

x'  =  <l>(x,y,  z,p,  q),    y' =  <l>(x,y,z,p,q),    z' = -•-,    /=•••,    ?'=•••, 
as  will  transform  into  itself  the  linear  differential  equation 
dz  —pdx  —  qdy  =  o. 

The  geometrical  meaning  of  the  transformation  is  evidently  that 
any  Mz  having  the  given  property  is  changed  into  an  M^  having 
the  same  property.  Thus,  for  instance,  a  surface  is  transformed 
generally  into  a  surface,  or  in  special  cases  into  a  point  or  a 
curve.  Moreover,  let  us  consider  two  manifoldnesses  M2  having 
a  contact,  i.e.  having  a  surface-element  in  common  ;  these  Mz's 
are  changed  by  the  transformation  into  two  other  M^s>  having 


SOPHUS   LIE.  21 

also  a  contact.  From  this  characteristic  the  name  given  by 
Lie  to  the  transformation  will  be  understood. 

Contact-transformations  are  so  important,  and  occur  so  fre- 
quently, that  particular  cases  attracted  the  attention  of  geome- 
ters long  ago,  though  not  under  this  name  and  from  this  point 
of  view,  i.e.  not  as  contact-transformations,  so  that  the  true 
insight  into  their  nature  could  not  be  obtained. 

Numerous  examples  of  contact-transformations  are  given 
in  my  (lithographed)  lectures  on  Hohere  Geometrie,  delivered 
during  the  winter-semester  of  1892-93.  Thus,  an  example 
in  two  dimensions  is  found  in  the  problem  of  wheel-gearing. 
The  outline  of  the  tooth  of  one  wheel  being  given,  it  is  here 
required  to  find  the  outline  of  the  tooth  of  the  other  wheel, 
as  I  explained  to  you  in  my  lecture  at  the  Chicago  Exhibition, 
with  the  aid  of  the  models  in  the  German  university  exhibit. 

Another  example  is  found  in  the  theory  of  perturbations  in 
astronomy ;  Lagrange's  method  of  variation  of  parameters  as 
applied  to  the  problem  of  three  bodies  is  equivalent  to  a 
contact-transformation  in  a  higher  space. 

The  group  of  oo15  substitutions  considered  yesterday  in 
line-geometry  is  also  a  group  of  contact-transformations,  both 
the  collineations  and  reciprocations  having  this  character. 
The  reciprocations  give  the  first  well-known  instance  of  the 
transformation  of  a  point  into  a  plane  (i.e.  a  surface),  and  a 
curve  into  a  developable  (i.e.  also  a  surface).  These  trans- 
formations of  curves  will  here  be  considered  as  transforming 
the  elements  of  the  points  or  curves  into  the  elements  of  the 
surface. 

Finally,  we  have  examples  of  contact-transformations,  not 
only  in  the  transformations  of  spheres  discussed  in  the  last 
lecture,  but  even  in  the  general  transition  from  the  line- 
geometry  of  Pliicker  to  the  sphere-geometry  of  Lie.  Let  us 
consider  this  last  case  somewhat  more  in  detail. 


22  LECTURE    III. 

First  of  all,  two  lines  that  intersect  have,  of  course,  a 
surface-element  in  common ;  and  as  the  two  corresponding 
spheres  must  also  have  a  surface-element  in  common,  they 
will  be  in  contact,  as  is  actually  the  case  for  our  transformation. 
It  will  be  of  interest  to  consider  more  closely  the  correlation 
between  the  surface-elements  of  a  line  and  those  of  a  sphere, 
although  it  is  given  by  imaginary  formulae.  Take,  for  instance, 
the  totality  of  the  surface-elements  belonging  to  a  circle  on 
one  of  the  spheres  ;  we  may  call  this  a  circular  set  of  elements. 
In  line-geometry  there  corresponds  the  set  of  surface-elements 
along  a  generating  line  of  a  skew  surface  ;  and  so  on.  The 
theorem  regarding  the  transformation  of  the  curves  of  curva- 
ture into  asymptotic  lines  becomes  now  self-evident.  Instead 
of  the  curve  of  curvature  of  a  surface  we  have  here  to  con- 
sider the  corresponding  elements  of  the  surface  which  we  may 
call  a  ctirvature  set.  Similarly,  an  asymptotic  line  is  replaced 
by  the  elements  of  the  surface  along  this  line ;  to  this  the  name 
osculating  set  may  be  given.  The  correspondence  between  the 
two  sets  is  brought  out  immediately  by  considering  that  two 
consecutive  elements  of  a  curvature  set  belong  to  the  same 
sphere,  while  two  consecutive  elements  of  an  osculating  set 
belong  to  the  same  straight  line. 

One  of  the  most  important  applications  of  contact-transforma- 
tions is  found  in  the  theory  of  partial  differential  equations ; 
I  shall  here  confine  myself  to  partial  differential  equations  of 
the  first  order.  From  our  new  point  of  view,  this  theory 
assumes  a  much  higher  degree  of  perspicuity,  and  the  true 
meaning  of  the  terms  "solution,"  "general  solution,"  "com- 
plete solution,"  "  singular  solution,"  introduced  by  Lagrange 
and  Monge,  is  brought  out  with  much  greater  clearness. 

Let  us  consider  the  partial  differential  equation  of  the  first 
order 

f(x,  y,  z,  p,  q)  =  o. 


SOPHUS   LIE.  23 

In  the  older  theory,  a  distinction  is  made  according  to  the  way 
in  which  /  and  q  enter  into  the  equation.  Thus,  when  /  and 
q  enter  only  in  the  first  degree,  the  equation  is  called  linear. 
If/  and  q  should  happen  to  be  both  absent,  the  equation  would 
not  be  regarded  as  a  differential  equation  at  all.  From  the 
higher  point  of  view  of  Lie's  new  geometry,  this  distinction 
disappears  entirely,  as  will  be  seen  in  what  follows. 

The  number  of  all  surface  elements  in  the  whole  of  space  is 
of  course  oo6.  By  writing  down  our  equation  we  single  out 
from  these  a  manifoldness  of  four  dimensions,  J/4,  of  oo4  ele- 
ments. Now,  to  find  a  "  solution "  of  the  equation  in  Lie's 
sense  means  to  single  out  from  this  M±  a  twofold  manifoldness, 
My  of  the  characteristic  property ;  whether  this  M%  be  a  point, 
a  curve,  or  a  surface,  is  here  regarded  as  indifferent.  What 
Lagrange  calls  finding  a  "complete  solution"  consists  in 
dividing  the  J/4  into  oo2  M2's.  This  can  of  course  be  done 
in  an  infinite  number  of  ways.  Finally,  if  any  singly  infinite 
set  be  taken  out  of  the  oo2  M2's,  we  have  in  the  envelope  of 
this  set  what  Lagrange  calls  a  "general  solution."  These 
formulations  hold  quite  generally  for  all  partial  differential 
equations  of  the  first  order,  even  for  the  most  specialized  forms. 

To  illustrate,  by  an  example,  in  what  sense  an  equation  of 
the  form  f(x,  y,  z)=Q  may  be  regarded  as  a  partial  differ- 
ential equation  and  what  is  the  meaning  of  its  solutions,  let 
us  consider  the  very  special  case  2  =  0.  While  in  ordinary 
co-ordinates  this  equation  represents  all  the  points  of  the  xy- 
plane,  in  Lie's  system  it  represents  of  course  all  the  surface- 
elements  whose  points  lie  in  the  plane.  Nothing  is  so  simple 
as  to  assign  a  "complete  solution"  in  this  case;  we  have  only 
to  take  the  oo2  points  of  the  plane  themselves,  each  point  being 
an  M2  of  the  equation.  To  derive  from  this  the  "  general  solu- 
tion," we  must  take  all  possible  singly  infinite  sets  of  points 
in  the  plane,  i.e.  any  curve  whatever,  and  form  the  envelope 


24  LECTURE   III. 

of  the  surface-elements  belonging  to  the  points  ;  in  other  words, 
we  must  take  the  elements  touching  the  curve.  Finally,  the 
plane  itself  represents  of  course  a  "singular  solution." 

Now,  the  very  high  interest  and  importance  of  this  simple 
illustration  lies  in  the  fact  that  by  a  contact-transformation 
every  partial  differential  equation  of  the  first  order  can  be 
changed  into  this  particular  form  z=o.  Hence  the  whole  dis- 
position of  the  solutions  outlined  above  holds  quite  generally. 

A  new  and  deeper  insight  is  thus  gained  through  Lie's 
theory  into  the  meaning  of  problems  that  have  long  been 
regarded  as  classical,  while  at  the  same  time  a  full  array  of 
new  problems  is  brought  to  light  and  finds  here  its  answer. 

It  can  here  only  be  briefly  mentioned  that  Lie  has  done  much 
in  applying  similar  principles  to  the  theory  of  partial  differential 
equations  of  the  second  order. 

At  the  present  time  Lie  is  best  known  through  his  theory  of 
continuous  groups  of  transformations,  and  at  first  glance  it 
might  appear  as  if  there  were  but  little  connection  between  this 
theory  and  the  geometrical  considerations  that  engaged  our 
attention  in  the  last  two  lectures.  I  think  it  therefore  desira- 
ble to  point  out  here  this  connection.  It  has  been  the  final 
aim  of  Lie  from  the  beginning  to  make  progress  in  the  theory 
of  differential  equations  ;  and  as  subsidiary  to  this  end  may  be 
regarded  both  the  geometrical  developments  considered  in  these 
lectures  and  the  theory  of  continuous  groups. 

For  further  particulars  concerning  the  subjects  of  the  present 
as  well  as  the  two  preceding  lectures,  I  may  refer  to  my  (litho- 
graphed) lectures  on  Hohere  Geometric,  delivered  at  Gottingen, 
in  1892-93.  The  theory  of  surface-elements  is  also  fully  devel- 
oped in  the  second  volume  of  the  Theorie  der  Transformations- 
gruppen,  by  Lie  and  Engel  (Leipzig,  Teubner,  1890). 


LECTURE    IV.  :     ON    THE    REAL    SHAPE    OF    ALGE- 
BRAIC  CURVES   AND    SURFACES. 

(August  31,  1893.) 

WE  turn  now  to  algebraic  functions,  and  in  particular  to  the 
question  of  the  actual  geometric  forms  corresponding  to  such 
functions.  The  question  as  to  the  reality  of  geometric  forms 
and  the  actual  shape  of  algebraic  curves  and  surfaces  was  some- 
what neglected  for  a  long  time.  Otherwise  it  would  be  difficult 
to  explain,  for  instance,  why  the  connection  between  Cayley's 
theory  of  projective  measurement  and  the  non-Euclidean  geom- 
etry should  not  have  been  perceived  at  once.  As  these  ques- 
tions are  even  now  less  well  known  than  they  deserve  to  be,  I 
proceed  to  give  here  an  historical  sketch  of  the  subject,  without, 
however,  attempting  completeness. 

It  must  be  counted  among  the  lasting  merits  of  Sir  Isaac 
Newton  that  he  first  investigated  the  shape  of  the  plane  curves 
of  the  third  order.  His  Enumeratio  linearum  tertii  ordinis* 
shows  that  he  had  a  very  clear  conception  of  projective 
geometry ;  for  he  says  that  all  curves  of  the  third  order  can 
be  derived  by  central  projection  from  five  fundamental  types 
(Fig.  i).  But  I  wish  to  direct  your  particular  attention  to  the 
paper  by  Mobius,  Ueber  die  Grundformen  der  Linien  der  dritten 
Ordnung,^  where  the  forms  of  the  cubic  curves  are  derived  by 

*  First  published  as  an  appendix  to  Newton's  Opticks,  1 704. 

f  Abhandlungen  der  Konigl.  Sachsischen  Gesellschaft  der  Wissenschaften,  math.- 
phys.  Klasse,  Vol.  I  (1852),  pp.  1-82;  reprinted  in  Mobius'  Gesammelte  Werke, 
Vol.  II  (1886),  pp.  89-176. 


26 


LECTURE   IV. 


purely  geometric  considerations.  Owing  to  its  remarkable 
elegance  of  treatment,  this  paper  has  given  the  impulse  to 
all  the  subsequent  researches  in  this  line  that  I  shall  have 
to  mention. 

In  1872  we  considered,  in  Gottingen,  the  question  as  to  the 
shape  of  surfaces  of  the  third  order.  As  a  particular  case, 
Clebsch  at  this  time  constructed  his  beautiful  model  of  the 


Fig.  1. 

diagonal  surface,  with  27  real  lines,  which  I  showed  to  you  at 
the  Exhibition.  The  equation  of  this  surface  may  be  written 
in  the  simple  form 


=  o 


which  shows  that  the  surface  can  be  transformed  into  itself  by 
the  1  20  permutations  of  the  xs. 

It  may  here  be  mentioned  as  a  general  rule,  that  in  select- 
ing a  particular  case  for  constructing  a  model  the  first  pre- 
requisite is  regularity.  By  selecting  a  symmetrical  form  for 
the  model,  not  only  is  the  execution  simplified,  but  what  is  of 
more  importance,  the  model  will  be  of  such  a  character  as  to 
impress  itself  readily  on  the  mind. 

Instigated  by  this  investigation  of  Clebsch,  I  turned  to  the 
general  problem  of  determining  all  possible  forms  of  cubic  sur- 


ALGEBRAIC  CURVES  AND  SURFACES.        27 

faces.*  I  established  the  fact  that  by  the  principle  of  continu- 
ity all  forms  of  real  surfaces  of  the  third  order  can  be  derived 
from  the  particular  surface  having  four  real  conical  points. 
This  surface,  also,  I  exhibited  to  you  at  the  World's  Fair,  and 
pointed  out  how  the  diagonal  surface  can  be  derived  from  it. 
But  what  is  of  primary  importance  is  the  completeness  of 
enumeration  resulting  from  my  point  of  view ;  it  would  be  of 
comparatively  little  value  to  derive  any  number  of  special  forms 
if  it  cannot  be  proved  that  the  method  used  exhausts  the 
subject.  Models  of  the  typical  cases  of  all  the  principal  forms 
of  cubic  surfaces  have  since  been  constructed  by  Rodenberg  for 
Brill's  collection. 

In  the  7th  volume  of  the  Math.  Annalen  (1874)  Zeuthenf  has 
discussed  the  various  forms  of  plane  curves  of  the  fourth  order 
(Q).  He  considers  in  particular  the  reality 
of  the  double  tangents  on  these  curves.  The 
number  of  such  tangents  is  28,  and  they  are 
all  real  when  the  curve  consists  of  four  sepa- 
rate closed  portions  (Fig.  2).  What  is  of  par- 
ticular interest  is  the  relation  of  Zeuthen's 
researches  on  quartic  curves  to  my  own  re-  p.  _ 

searches  on  cubic  surfaces,  as  explained  by 
Zeuthen  himself.}  It  had  been  observed  before,  by  Geiser,  that 
if  a  cubic  surface  be  projected  on  a  plane  from  a  point  on  the 
surface,  the  contour  of  the  projection  is  a  quartic  curve,  and 
that  every  quartic  curve  can  be  generated  in  this  way.  If  a 
surface  with  four  conical  points  be  chosen,  the  resulting  quartic 
has  four  double  points  ;  that  is,  it  breaks  up  into  two  conies 

*  See  my  paper  Ueber  Flachen  driller  Ordnung,  Math.  Annalen,  Vol.  6  (1873), 
pp.  551-581. 

t  Sur  les  differenles  formes  des  courbes  planes  du  qualrieme  ordre,  pp.  410-432. 

J  £tudes  des  proprieles  de  situation  des  surfaces  cubiques,  Math.  Annalen,  Vol.  8 
(1875),  pp.  1-30. 


d  D 


28  LECTURE   IV. 

(Fig.  3).  By  considering  the  shaded  portions  in  the  figure  it 
will  readily  be  seen  how,  by  the  principle  of  continuity,  the  four 
ovals  of  the  quartic  (Fig.  2)  are  obtained.  This  corresponds 
exactly  to  the  derivation  of  the  diagonal 
surface  from  the  cubic  surface  having  four 
conical  points. 

The  attempts  to  extend  this  application 
of  the  principle  of  continuity  so  as  to  gain 
an  insight  into  the  shape  of  curves  of  the 
nth  order  have  hitherto  proved  futile,  as 
far  as  a  general  classification  and  an  enu- 
meration of  all  fundamental  forms  is  concerned.  Still,  some 
important  results  have  been  obtained.  A  paper  by  Harnack* 
and  a  more  recent  one  by  Hilbertf  are  here  to  be  mentioned. 
Harnack  finds  that,  if  /  be  the  deficiency  of  the  curve,  the 
maximum  number  of  separate  branches  the  curve  can  have  is 
/+i;  and  a  curve  with  p+i  branches  actually  exists.  Hil- 
bert's  paper  contains  a  large  number  of  interesting  special 
results  which  from  their  nature  cannot  be  included  in  the 
present  brief  summary. 

I  myself  have  found  a  curious  relation  between  the  numbers 
of  real  singularities.^:  Denoting  the  order  of  the  curve  by  «, 
the  class  by  k,  and  considering  only  simple  singularities,  we 
may  have  three  kinds  of  double  points,  say  d'  ordinary  and  d" 
isolated  real  double  points,  besides  imaginary  double  points  ; 
then  there  may  be  r1  real  cusps,  besides  imaginary  cusps ;  and 
similarly,  by  the  principle  of  duality,  /'  ordinary,  /"  isolated 


*  Ueber  die  Vieltheiligkeit  der  ebcnen  algebraischen  Curven,  Math.  Annalen,  Vol. 
10  (1876),  pp.  189-198. 

f  Ueber  die  reellen  ZUge  algebraischer  Curven,  Math.  Annalen,  Vol.  38  (1891), 
pp.  115-138. 

J  Eine  neue  Relation  zwischen  den  Singularitdten  einer  algebraischen  Curve, 
Math.  Annalen,  Vol.  10  (1876),  pp.  199-209. 


ALGEBRAIC  CURVES  AND  SURFACES.        29 

real  double  tangents,  besides  imaginary  double  tangents ;  also 
w'  real  inflexions,  besides  imaginary  inflexions.  Then  it  can 
be  proved  by  means  of  the  principle  of  continuity,  that  the 
following  relation  must  hold  : 

n  +  w'  +  2 1"  =  k  +  r1  +  2  d". 

This  general  law  contains  everything  that  is  known  as  to 
curves  of  the  third  or  fourth  orders.  It  has  been  somewhat 
extended  in  a  more  algebraic  sense  by  several  writers.  More- 
over, Brill,  in  Vol.  16  of  the  Math.  Annalen  (1880),*  has  shown 
how  the  formula  must  be  modified  when  higher  singularities  are 
involved. 

As  regards  quartic  surfaces,  Rohn  has  investigated  an  enor- 
mous number  of  special  cases  ;  but  a  complete  enumeration  he 
has  not  reached.  Among  the  special 
surfaces  of  the  fourth  order  the  Kum- 
mer  surface  with  16  conical  points  is 
one  of  the  most  important.  The 
models  constructed  by  Plucker  in 
connection  with  his  theory  of  com- 
plexes of  lines  all  represent  special 
cases  of  the  Kummer  surface.  Some 
types  of  this  surface  are  also  included 
in  the  Brill  collection.  But  all  these 
models  are  now  of  less  importance, 
since  Rohn  found  the  following  in- 
teresting and  comprehensive  result. 


Fig.  4. 


Imagine  a  quadric  surface  with  four  generating  lines  of  each  set 
(Fig.  4).  According  to  the  character  of  the  surface  and  the 
reality,  non-reality,  or  coincidence  of  these  lines,  a  large  number 
of  special  cases  is  possible  ;  all  these  cases,  however,  must  be 


*  Ueber  Singular  it aten  ebener  algebraischer  Curven  und  eine  neue  Curvenspecies, 
pp.  348-408. 


30  LECTURE   IV. 

treated  alike.  We  may  here  confine  ourselves  to  the  case  of 
an  hyperboloid  of  one  sheet  with  four  distinct  lines  of  each 
set.  These  lines  divide  the  surface  into  16  regions.  Shading 
the  alternate  regions  as  in  the  figure,  and  regarding  the  shaded 
regions  as  double,  the  unshaded  regions  being  disregarded,  we 
have  a  surface  consisting  of  eight  separate  closed  portions  hang- 
ing together  only  at  the  points  of  intersection  of  the  lines  ;  and 
this  is  a  Kummer  surface  with  16  real  double  points.  Rohn's 
researches  on  the  Kummer  surface  will  be  found  in  the  Math. 
Annalen,  Vol.  18  (1881);*  his  more  general  investigations  on 
quartic  surfaces,  ib.,  Vol.  29  (1887)-! 

There  is  still  another  mode  of  dealing  with  the  shape  of 
curves  (not  of  surfaces),  viz.  by  means  of  the  theory  of  Rie- 
mann. The  first  problem  that  here  presents  itself  is  to  estab- 
lish the  connection  between  a  plane  curve  and  a  Riemann  sur- 
face, as  I  have  done  in  Vol.  7  of  the  Math.  Annalen  (1874)4 
Let  us  consider  a  cubic  curve  ;  its  deficiency  is  /=  I.  Now  it 
is  well  known  that  in  Riemann's  theory  this  deficiency  is  a 
measure  of  the  connectivity  of  the  corresponding  Riemann  sur- 
face, which,  therefore,  in  the  present  case,  must  be  that  of  a 
torus,  or  anchor-ring.  The  question  then  arises :  what  has  the 
anchor-ring  to  do  with  the  cubic  curve  ?  The  connection  will 
best  be  understood  by  considering  the  curve  of  the  third  class 
whose  shape  is  represented  in  Fig.  5.  It  is  easy  to  see  that  of 
the  three  tangents  that  can  be  drawn  to  this  curve  from  any 
point  in  its  plane,  all  three  will  be  real  if  the  point  be  selected 
outside  the  oval  branch,  or  inside  the  triangular  branch  ;  but  that 
only  one  of  the  three  tangents  will  be  real  for  any  point  in  the 
shaded  region,  while  the  other  two  tangents  are  imaginary.  As 

*  Die  verschiedenen  Gestalten  der  Kummer 'schen  Flache,  pp.  99-159. 
t  Die  Flachen  vierter  Ordnung  hinsichtlich  ihrer  Knotenpunkte  und  ihrer  Ge- 
staltung,  pp.  81-96. 

\  Ueber  eine  neue  Art  der  Riemann'' schen  Flachen,  pp.  558—566. 


ALGEBRAIC  CURVES  AND  SURFACES.        31 

there  are  thus  two  imaginary  tangents  corresponding  to  each 
point  of  this  region,  let  us  imagine  it  covered  with  a  double 
leaf ;  along  the  curve  the  two  leaves  must,  of  course,  be 
regarded  as  joined.  Thus  we  obtain  a  surface  which  can  be 
considered  as  a  Riemann  surface  belonging  to  the  curve,  each 
point  of  the  surface  corresponding  to  a  single  tangent  of  the 
curve.  Here,  then,  we  have  our  anchor-ring.  If  on  such  a  sur- 
face we  study  integrals,  they  will  be  of  double  periodicity,  and 
the  true  reason  is  thus  disclosed  for  the  connection  of  elliptic 


Fig.  5. 


integrals  with  the  curves  of  the  third  class,  and  hence,  owing 
to  the  relation  of  duality,  with  the  curves  of  the  third  order. 

To  make  a  further  advance,  I  passed  to  the  general  theory 
of  Riemann  surfaces.  To  real  curves  will  of  course  correspond 
symmetrical  Riemann  surfaces,  i.e.  surfaces  that  reproduce 
themselves  by  a  conformal  transformation  of  the  second  kind 
(i.e.  a  transformation  that  inverts  the  sense  of  the  angles). 
Now  it  is  easy  to  enumerate  the  different  symmetrical  types 
belonging  to  a  given  /.  The  result  is  that  there  are  altogether 

/>+i    " diasymmetric  "    and  \—         " orthosymmetric  "   cases. 
If  we  denote  as  a  line  of  symmetry   any  line  whose  points 


32  LECTURE   IV. 

remain  unchanged  by  the  conformal  transformation,  the  dia- 
symmetric  cases  contain  respectively  /,/—!,•••  2,  I,  o  lines 
of  symmetry,  and  the  orthosymmetric  cases  contain  /+  I,  /—  i, 
/  — 3)-"  such  lines.  A  surface  is  called  diasymmetric  or  ortho- 
symmetric  according  as  it  does  not  or  does  break  up  into  two 
parts  by  cuts  carried  along  all  the  lines  of  symmetry.  This 
enumeration,  then,  will  contain  a  general  classification  of  real 
curves,  as  indicated  first  in  my  pamphlet  on  Riemann's  theory.* 
In  the  summer  of  1892  I  resumed  the  theory  and  developed 
a  large  number  of  propositions  concerning  the  reality  of  the 
roots  of  those  equations  connected  with  our  curves  that  can  be 
treated  by  means  of  the  Abelian  integrals.  Compare  the  last 
volume  of  the  Math.  Annalen^  and  my  (lithographed)  lectures 
on  Riemanrische  Flachen,  Part  II. 

In  the  same  manner  in  which  we  have  to-day  considered 
ordinary  algebraic  curves  and  surfaces,  it  would  be  interesting 
to  investigate  all  algebraic  configurations  so  as  to  arrive  at  a 
truly  geometrical  intuition  of  these  objects. 

In  concluding,  I  wish  to  insist  in  particular  on  what  I  regard 
as  the  principal  characteristic  of  the  geometrical  methods  that  I 
have  discussed  to-day :  these  methods  give  us  an  actual  mental 
image  of  the  configuration  under  discussion,  and  this  I  consider 
as  most  essential  in  all  true  geometry.  For  this  reason  the 
so-called  synthetic  methods,  as  usually  developed,  do  not  appear 
to  me  very  satisfactory.  While  giving  elaborate  constructions 
for  special  cases  and  details  they  fail  entirely  to  afford  a  general 
view  of  the  configurations  as  a  whole. 


*  Ueber  Riemanri's  Theorie  der  algebraischen  Functionen  und  ihrer  Integrate, 
Leipzig,  Teubner,  1882.  An  English  translation  by  Frances  Hardcastle  (London, 
Macmillan)  has  just  appeared. 

f  Ueber  Realitatsverhaltnisse  bei  der  einem  beliebigen  Geschlechte  zugehorigcn 
Normalcurve  der  <f>,  Vol.  42  (1893),  pp.  1-29. 


LECTURE  V. :    THEORY   OF   FUNCTIONS   AND 
GEOMETRY. 

(September  i,  1893.) 

A  GEOMETRICAL  representation  of  a  function  of  a  complex 
variable  w=f(z),  where  w  =  u  +  iv  and  z=x  +  iy,  can  be  ob- 
tained by  constructing  models  of  the  two  surfaces  u  =  <j>(x,y), 
v=*ty-(x,y).  This  idea  is  realized  in  the  models  constructed 
by  Dyck,  which  I  have  shown  to  you  at  the  Exhibition. 

Another  well-known  method,  proposed  by  Riemann,  consists 
in  representing  each  of  the  two  complex  variables  in  the  usual 
way  in  a  plane.  To  every  point  in  the  ^-plane  will  correspond 
one  or  more  points  in  the  w-plane ;  as  z  moves  in  its  plane,  w 
describes  a  corresponding  curve  in  the  other  plane.  I  may 
refer  to  the  work  of  Holzmiiller*  as  a  good  elementary  intro- 
duction to  this  subject,  especially  on  account  of  the  large 
number  of  special  cases  there  worked  out  and  illustrated  by 
drawings. 

In  higher  investigations,  what  is  of  interest  is  not  so  much 
the  corresponding  curves  as  corresponding  areas  or  regions 
of  the  two  planes.  According  to  Riemann's  fundamental 
theorem  concerning  conformal  representation,  two  simply  con- 
nected regions  can  always  be  made  to  correspond  to  each  other 
conformally,  so  that  either  is  the  conformal  representation 

*  Einfuhrung  in  die  Theorie  der  isogonalen  Verwandtschaften  und  der  conformen 
Abbildungen,  verbunden  mit  Anwendungen  auf  mathematische  Physik,  Leipzig, 
Teubner,  1882. 

33 


34 


LECTURE   V. 


(Abbildung)  of  the  other.  The  three  constants  at  our  disposal 
in  this  correspondence  allow  us  to  select  three  arbitrary  points 
on  the  boundary  of  one  region  as  corresponding  to  three  arbi- 
trary points  on  the  boundary  of  the  other  region.  Thus 
Riemann's  theory  affords  a  geometrical  definition  for  any  func- 
tion whatever  by  means  of  its  conformal  representation. 

This  suggests  the  inquiry  as  to  what  conclusions  can  be 
drawn  from  this  method  concerning  the  nature  of  transcen- 
dental functions.  Next  to  the  elementary  transcendental  func- 
tions the  elliptic  functions  are  usually  regarded  as  the  most 
important.  There  is,  however,  another  class  for  which  at 
least  equal  importance  must  be  claimed  on  account  of  their 
numerous  applications  in  astronomy  and  mathematical  physics  ; 
these  are  the  hypergeometric  functions,  so  called  owing  to  their 
connection  with  Gauss's  hypergeometric  series. 

The  hypergeometric  functions  can  be  defined  as  the  integrals 
of  the  following  linear  differential  equation  of  the  second  order: 

i-X'-X",  v 


w 


z  —  b  z—c  \(z—a)(z—b}(z—c) 

where  2= a,  b,  c  are  the  three  singular  points  and  X',  X" ;  ///,  /A"  ; 
v',  v"  are  the  so-called  exponents  belonging  respectively  to 
a,  b,  c. 

If  W-L  be  a  particular  solution,  w2  another,  the  general  solution 
can  be  put  in  the  form  aw^^w^  where  a,  $  are  arbitrary  con- 
stants ;  so  that 

and  YI 


represent  a  pair  of  general  solutions. 


THEORY   OF   FUNCTIONS   AND   GEOMETRY. 


35 


If  we  now  introduce  the  quotient  ^  =  77  (2)  as  a  new  variable, 


^vft 


its  most  general  value  is 


and  contains  there- 


fore  three  arbitrary  constants.     Hence  77  satisfies  a  differential 
equation  of  the  third  order  which  is  readily  found  to  be 


"i-X* 


-  a) 


in  which  the  left-hand  member  has  the  property  of  not  being 
changed  by  a  linear  substitution,  and  is  therefore  called  a  differ- 
ential invariant.  Cayley  has  named  this  function  the  Schwar- 
zian  derivative ;  it  has  formed  the  starting-point  for  Sylvester's 
investigations  on  reciprocants.  In  the  right-hand  member, 

±X  =  X'-X",  ±/t  =  /*'-/*",  ±v  =  v'-v". 

As  to  the  conformal  representation  (Fig.  6),  it  can  be  shown 
that  the  upper  half  of  the  ^-plane,  with  the  points  a,  b,  c  on 


Fig.  6. 


the  real  axis  and  X,  /*,  v  assumed  as  real,  is  transformed  for  each 
branch  of  the  ^-function  into  a  triangular  area  abc  bounded  by 


36  LECTURE   V. 

three  circular  arcs ;  let  us  call  such  an  area  a  circular  triangle 
(Kreisbogendreieck}.  The  angles  at  the  vertices  of  this  triangle 
are  \TT,  /ATT,  VTT. 

This,  then,  is  the  geometrical  representation  we  have  to 
take  as  our  basis.  In  order  to  derive  from  it  conclusions  as 
to  the  nature  of  the  transcendental  functions  defined  by  the 
differential  equation,  it  will  evidently  be  necessary  to  inquire 
what  are  the  forms  of  such  circular  triangles  in  the  most 
general  case.  For  it  is  to  be  noticed  that  there  is  no  restric- 
tion laid  upon  the  values  of  the  constants  \,  ft,  v,  so  that  the 
angles  of  our  triangle  are  not  necessarily  acute,  nor  even 
convex ;  in  other  words,  in  the  general  case  the  vertices  will- 
be  branch-points.  The  triangle  itself  is  here  to  be  regarded 
as  something  like  an  extensible  and  flexible  membrane  spread 
out  between  the  circles  forming  the  boundary. 

I  have  investigated  this  question  in  a  paper  published  in 
the  Math.  Annalen,  Vol.  37.*  It  will  be  convenient  to  project 
the  plane  containing  the  circular  triangle  stereographically  on 
a  sphere.  The  question  then  is  as  to  the  most  general  form 
of  spherical  triangles,  taking  this  term  in  a  generalized  meaning 
as  denoting  any  triangle  on  the  sphere  bounded  by  the  inter- 
sections of  three  planes  with  the  sphere,  whether  the  planes 
intersect  at  the  centre  or  not. 

This  is  really  a  question  of  elementary  geometry ;  and  it  is 
interesting  to  notice  how  often  in  recent  times  higher  re- 
search has  led  back  to  elementary  problems  not  previously 
settled. 

The  result  in  the  present  case  is  that  there  are  two,  and 
only  two,  species  of  such  generalized  triangles.  They  are 
obtained  from  the  so-called  elementary  triangle  by  two  distinct 
operations  :  (a)  lateral,  (b)  polar  attachment  of  a  circle. 

*  Ueber  die  Nullstellen  df.r  hypergeometrischen  Reihe,  pp.  573-590. 


THEORY  OF  FUNCTIONS  AND  GEOMETRY. 


37 


Let  abc  (Fig.  7)  be  the  elementary  spherical  triangle.  Then 
the  operation  of  lateral  attachment  consists  in  attaching  to 
the  area  abc  the  area  enclosed  by  one  of  the  sides,  say  be, 
this  side  being  produced  so  as  to  form  a  complete  circle. 
The  process  can,  of  course,  be  repeated  any  number  of  times 
and  applied  to  each  side.  If  one  circular  area  be  attached  at 
be,  the  angles  at  b  and  c  are  increased  each  by  TT  ;  if  the 
whole  sphere  be  attached,  by  2ir,  etc.  The  vertices  in  this 
way  become  branch-points.  A  triangle  so  obtained  I  call  a 
triangle  of  the  first  species. 


Fig.  7. 


A  triangle  of  the  second  species  is  produced  by  the  process 
of  polar  attachment  of  a  circle,  say  at  bc\  the  whole  area 
bounded  by  the  circle  be  is,  in  this  case,  connected  with  the 
original  triangle  along  a  branch-cut  reaching  from  the  vertex 
a  to  some  point  on  be.  The  point  a  becomes  a  branch-point, 
its  angle  being  increased  by  2?r.  Moreover,  lateral  attach- 
ments can  be  made  at  ab  and  ac. 

The  two  species  of  triangles  are  now  characterized  as  follows : 
the  first  species  may  have  any  number  of  lateral  attachments 
at  any  or  all  of  the  three  sides,  while  the  second  has  a  polar 
attachment  to  one  vertex  and  the  opposite  side,  and  may  have 
lateral  attachments  to  the  other  two  sides. 


38  LECTURE   V. 

Analytically  the  two  species  are  distinguished  by  inequali- 
ties between  the  absolute  values  of  the  constants  \,  p,  v.  For 
the  first  species,  none  of  the  three  constants  is  greater  than 
the  sum  of  the  other  two,  i.e. 


for  the  second  species, 


where  A,  refers  to  the  pole. 

For  the  application  to  the  theory  of  functions,  it  is  impor- 
tant to  determine,  in  the  case  of  the  second  species,  the 
number  of  times  the  circle  formed  by  the  side  opposite  the 
vertex  is  passed  around.  I  have  found  this  number  to  be 

E  (-  -),  where  E  denotes  the  greatest  positive 

integer  contained  in  the  argument,  and  is  therefore  always  zero 
when  this  argument  happens  to  be  negative  or  fractional. 

Let  us  now  apply  these  geometrical  ideas  to  the  theory  of 
hypergeometric  functions.  I  can  here  only  point  out  one  of 
the  results  obtained.  Considering  only  the  real  values  that 
?7=a/1/w2  can  assume  between  a  and  b,  the  question  presents 
itself  as  to  the  shape  of  the  i;-curve  between  these  limits. 
Let  us  consider  for  a  moment  the  curves  w-^  and  w2.  It  is 
well  known  that,  if  w±  oscillates  between  a  and  b  from  one 
side  of  the  axis  to  the  other,  w2  will  also  oscillate  ;  their 
quotient  rj  =  zv1/zvz  is  represented  by  a  curve  that  consists  of 
separate  branches  extending  from  —  oo  to  +00,  somewhat  like 
the  curve  j/  =  tan.r.  Now  it  appears  as  the  result  of  the 
investigation  that  the  number  of  these  branches,  and  therefore 
the  number  of  the  oscillations  of  zt>l  and  w2,  is  given  precisely 
by  the  number  of  circuits  of  the  point  c  ;  that  is  to  say,  it  is 

E\~  -}•     This  is  a  result  of  importance  for  all 

V  2  / 


THEORY    OF   FUNCTIONS   AND   GEOMETRY. 


39 


applications  of  hypergeometric  functions  which  was  derived 
only  later  (by  Hurwitz)  by  means  of  Sturm's  methods. 

I  wish  to  call  your  particular  attention  not  so  much  to  the 
result  itself,  however  interesting  it  may  be,  as  to  the  geometrical 
method  adopted  in  deriving  it.  More  advanced  researches  on  a 
similar  line  of  thought  are  now  being  carried  on  at  Gottingen 
by  myself  and  others. 

When  a  differential  equation  with  a  larger  number  of  singular 
points  than  three  is  the  object  of  investigation,  the  triangles 
must  be  replaced  by  quadrangles  and  other  polygons.  In  my 
lithographed  lectures  on  Linear  Differential  Equations,  delivered 
in  1890-91,  I  have  thrown  out  some  suggestions  regarding 
the  treatment  of  such  cases.  The  difficulty  arising  in  these 
generalizations  is,  strange  to  say,  merely  of  a  geometrical 
nature,  viz.  the  difficulty  of  obtaining  a  general  view  of  the 
possible  forms  of  the  polygons. 

Meanwhile,  Dr.  Schoenflies  has  published  a  paper  on  recti- 
linear polygons  of  any  number  of  sides*  while  Dr.  Van  Vleck 
has  considered  such  rectilinear  polygons  together  with  the 
functions  they  define,  the  polygons  being  defined  in  so  general 
a  way  as  to  admit  branch-points  even  in  the  interior.  Dr. 
Schoenflies  has  also  treated  the  case  of  circular  quadrangles, 
the  result  being  somewhat  complicated. 

In  all  these  investigations  the  singular  points  of  the  ^-plane 
corresponding  to  the  vertices  of  the  polygons  are  of  course 
assumed  to  be  real,  as  are  also  their  exponents.  There  remains 
the  still  more  general  question  how  to  represent  by  conformal 
correspondence  the  functions  in  the  case  when  some  of  these 
elements  are  complex.  In  this  direction  I  have  to  mention  the 
name  of  Dr.  Schilling  who  has  treated  the  case  of  the  ordinary 
hypergeometric  function  on  the  assumption  of  complex  exponents. 

*  Ueber  Kreisbogenpolygone,  Math.  Annalen,  Vol.  42,  pp.  377-408. 


40  LECTURE   V. 

This  treatment  of  the  functions  defined  by  linear  differential 
equations  of  the  second  order  is  of  course  only  an  example 
of  the  general  discussion  of  complex  functions  by  means  of 
geometry.  I  hope  that  many  more  interesting  results  will  be 
obtained  in  the  future  by  such  geometrical  methods. 


LECTURE  VI.:  ON  THE  MATHEMATICAL  CHAR- 
ACTER OF  SPACE-INTUITION  AND  THE 
RELATION  OF  PURE  MATHEMATICS  TO 
THE  APPLIED  SCIENCES. 

(September  2,  1893.) 

IN  the  preceding  lectures  I  have  laid  so  much  stress  on 
geometrical  methods  that  the  inquiry  naturally  presents  itself 
as  to  the  real  nature  and  limitations  of  geometrical  intuition. 

In  my  address  before  the  Congress  of  Mathematics  at  Chi- 
cago I  referred  to  the  distinction  between  what  I  called  the 
naive  and  the  refined  intuition.  It  is  the  latter  that  we  find  in 
Euclid ;  he  carefully  develops  his  system  on  the  basis  of  well- 
formulated  axioms,  is  fully  conscious  of  the  necessity  of  exact 
proofs,  clearly  distinguishes  between  the  commensurable  and 
incommensurable,  and  so  forth. 

The  na'fve  intuition,  on  the  other  hand,  was  especially  active 
during  the  period  of  the  genesis  of  the  differential  and  integral 
calculus.  Thus  we  see  that  Newton  assumes  without  hesitation 
the  existence,  in  every  case,  of  a  velocity  in  a  moving  point, 
without  troubling  himself  with  the  inquiry  whether  there  might 
not  be  continuous  functions  having  no  derivative. 

At  the  present  time  we  are  wont  to  build  up  the  infinitesi- 
mal calculus  on  a  purely  analytical  basis,  and  this  shows  that 
we  a*re  living  in  a  critical  period  similar  to  that  of  Euclid. 
It  is  my  private  conviction,  although  I  may  perhaps  not  be 
able  to  fully  substantiate  it  with  complete  proofs,  that  Euclid's 

41 


42 


LECTURE   VI. 


period  also  must  have  been  preceded  by  a  "naive"  stage  of 
development.  Several  facts  that  have  become  known  only 
quite  recently  point  in  this  direction.  Thus  it  is  now  known 
that  the  books  that  have  come  down  to  us  from  the  time  of 
Euclid  constitute  only  a  very  small  part  of  what  was  then 
in  existence ;  moreover,  much  of  the  teaching  was  done  by 
oral  tradition.  Not  many  of  the  books  had  that  artistic  finish 
that  we  admire  in  Euclid's  "  Elements "  ;  the  majority  were 
in  the  form  of  improvised  lectures,  written  out  for  the  use 
of  the  students.  The  investigations  of  Zeuthen*  and  Allman  f 
have  done  much  to  clear  up  these  historical  conditions. 

If  we  now  ask  how  we  can  account  for  this  distinction 
between  the  na'fve  and  refined  intuition,  I  must  say  that,  in 
my  opinion,  the  root  of  the  matter  lies  in  the  fact  that  the 
naive  intuition  is  not  exact,  while  the  refined  intuition  is  not 
properly  intuition  at  all,  but  arises  through  the  logical  develop- 
ment from  axioms  considered  as  perfectly  exact. 

To  explain  the  meaning  of  the  first  half  of  this  statement  it 
is  my  opinion  that,  in  our  na'fve  intuition,  when  thinking  of 
a  point  we  do  not  picture  to  our  mind  an  abstract  mathemati- 
cal point,  but  substitute  something  concrete  for  it.  In  imagin- 
ing a  line,  we  do  not  picture  to  ourselves  "length  without 

breadth,"  but  a  strip  of  a  certain  width. 

Now  such  a  strip  has  of  course  always 
x        a  tangent  (Fig.  9) ;  i.e.  we  can  always 

imagine  a  straight  strip  having  a  small 

portion  (element)  in  common  with  the  curved  strip ;  similarly 
with  respect  to  the  osculating  circle.  The  definitions  in  this 
case  are  regarded  as  holding  only  approximately,  or  as  far  as 
may  be  necessary. 

*  Die  Lehre  von  den  Kegelschnitten  im  Altertum,  ubersetzt  von  R.  v.  Fischer- 
Benzon,  Kopenhagen,  Host,  1886. 

f  Greek  geometry  from  Thales  to  Euclid,  Dublin,  Hodges,  1889. 


MATHEMATICAL  CHARACTER  OF  SPACE-INTUITION. 


43 


The  "  exact "  mathematicians  will  of  course  say  that  such 
definitions  are  not  definitions  at  all.  But  I  maintain  that  in 
ordinary  life  we  actually  operate  with  such  inexact  definitions. 
Thus  we  speak  without  hesitancy  of  the  direction  and  curvature 
of  a  river  or  a  road,  although  the  "  line"  in  this  case  has  certainly 
considerable  width. 

As  regards  the  second  half  of  my  proposition,  there  actually 
are  many  cases  where  the  conclusions  derived  by  purely  logical 
reasoning  from  exact  definitions  can  no  more  be  verified  by 
intuition.  To  show  this,  I  select  examples  from  the  theory  of 
automorphic  functions,  because  in  more  common  geometrical 
illustrations  our  judgment  is  warped  by  the  familiarity  of  the 
ideas. 

Let  any  number  of  non-intersecting  circles  I,  2,  3,  4,  •••,  be 
given  (Fig.  10),  and  let  every  circle  be  reflected  (i.e.  transformed 


Fig.  10. 

by  inversion,  or  reciprocal  radii  vectores)  upon  every  other  circle ; 
then  repeat  this  operation  again  and  again,  ad  infinitum.  The 
question  is,  what  will  be  the  configuration  formed  by  the  totality 


44  LECTURE  VI. 

of  all  the  circles,  and  in  particular  what  will  be  the  position  of 
the  limiting  points.  There  is  no  difficulty  in  answering  these 
questions  by  purely  logical  reasoning ;  but  the  imagination 
seems  to  fail  utterly  when  we  try  to  form  a  mental  image  of 
the  result. 

Again,  let  a  series  of  circles  be  given,  each  circle  touching  the 
following,  while  the  last  touches  the  first  (Fig.  1 1).  Every  circle 
is  now  reflected  upon  every  other  just  as  in  the  preceding  exam- 
ple, and  the  process  is  repeated  indefinitely.  The  special  case 
when  the  original  points  of  contact  happen  to  lie  on  a  circle 


Fig.  11. 

being  excluded,  it  can  be  shown  analytically  that  the  continuous 
curve  which  is  the  locus  of  all  the  points  of  contact  is  not  an 
analytic  curve.  The  points  of  contact  form  a  manifoldness  that 
is  everywhere  dense  on  the  curve  (in  the  sense  of  G.  Cantor), 
although  there  are  intermediate  points  between  them.  At 
each  of  the  former  points  there  is  a  determinate  tangent, 
while  there  is  none  at  the  intermediate  points.  Second  deriv- 
atives do  not  exist  at  all.  It  is  easy  enough  to  imagine  a  strip 
covering  all  these  points ;  but  when  the  width  of  the  strip  is 
reduced  beyond  a  certain  limit,  we  find  undulations,  and  it  seems 
impossible  to  clearly  picture  to  the  mind  the  final  outcome. 
It  is  to  be  noticed  that  we  have  here  an  example  of  a  curve 


MATHEMATICAL  CHARACTER  OF  SPACE-INTUITION.       45 

with  indeterminate  derivatives  arising  out  of  purely  geometrical 
considerations,  while  it  might  be  supposed  from  the  usual 
treatment  of  such  curves  that  they  can  only  be  defined  by 
artificial  analytical  series. 

Unfortunately,  I  am  not  in  a  position  to  give  a  full  account 
of  the  opinions  of  philosophers  on  this  subject.  As  regards 
the  more  recent  mathematical  literature,  I  have  presented  my 
views  as  developed  above  in  a  paper  published  in  1873,  and 
since  reprinted  in  the  Math.  Annalen*  Ideas  agreeing  in 
general  with  mine  have  been  expressed  by  Pasch,  of  Giessen, 
in  two  works,  one  on  the  foundations  of  geometry,!  the  other 
on  the  principles  of  the  infinitesimal  calculus.:}:  Another 
author,  Kopcke,  of  Hamburg,  has  advanced  the  idea  that  our 
space-intuition  is  exact  as  far  as  it  goes,  but  so  limited  as  to 
make  it  impossible  for  us  to  picture  to  ourselves  curves  with- 
out tangents.  § 

On  one  point  Pasch  does  not  agree  with  me,  and  that  is  as  to 
the  exact  value  of  the  axioms.  He  believes  — and  this  is  the 
traditional  view  —  that  it  is  possible  finally  to  discard  intuition 
entirely,  basing  the  whole  science  on  the  axioms  alone.  I  am 
of  the  opinion  that,  certainly,  for  the  purposes  of  research  it  is 
always  necessary  to  combine  the  intuition  with  the  axioms.  I 
do  not  believe,  for  instance,  that  it  would  have  been  possible  to 
derive  the  results  discussed  in  my  former  lectures,  the  splendid 
researches  of  Lie,  the  continuity  of  the  shape  of  algebraic  curves 
and  surfaces,  or  the  most  general  forms  of  triangles,  without 
the  constant  use  of  geometrical  intuition. 


*  Ueber  den  allgemeincn  Functionsbegriff  und  dessen  Darstellung  durch  tine 
•willkurliche  Curve,  Math.  Annalen,  Vol.  22  (1883),  pp.  249-259. 

f  Vorlesungen  iiber  neuere  Geomttrie,  Leipzig,  Teubner,  1882. 

J  Einleitung  in  die  Differential-  und  Integralrechnung,  Leipzig,  Teubner,  1882. 

§  Ueber  Differentiirbarkeit  und  Anschaulichkeit  der  stetigen  Functionen,  Math. 
Annalen,  Vol.  29  (1887),  pp.  123-140. 


46  LECTURE   VI. 

Pasch's  idea  of  building  up  the  science  purely  on  the  basis  of 
the  axioms  has  since  been  carried  still  farther  by  Peano,  in  his 
logical  calculus. 

Finally,  it  must  be  said  that  the  degree  of  exactness  of  the 
intuition  of  space  may  be  different  in  different  individuals,  per- 
haps even  in  different  races.  It  would  seem  as  if  a  strong 
naive  space-intuition  were  an  attribute  pre-eminently  of  the 
Teutonic  race,  while  the  critical,  purely  logical  sense  is  more 
fully  developed  in  the  Latin  and  Hebrew  races.  A  full  investi- 
gation of  this  subject,  somewhat  on  the  lines  suggested  by 
Francis  Galton  in  his  researches  on  heredity,  might  be  inter- 
esting. 

What  has  been  said  above  with  regard  to  geometry  ranges 
this  science  among  the  applied  sciences.  A  few  general 
remarks  on  these  sciences  and  their  relation  to  pure  mathe- 
matics will  here  not  be  out  of  place.  From  the  point  of  view 
of  pure  mathematical  science  I  should  lay  particular  stress  on 
the  heuristic  value  of  the  applied  sciences  as  an  aid  to  discov- 
ering new  truths  in  mathematics.  Thus  I  have  shown  (in  my 
little  book  on  Riemann's  theories)  that  the  Abelian  integrals 
can  best  be  understood  and  illustrated  by  considering  electric 
currents  on  closed  surfaces.  In  an  analogous  way,  theorems 
concerning  differential  equations  can  be  derived  from  the  con- 
sideration of  sound-vibrations ;  and  so  on. 

But  just  at  present  I  desire  to  speak  of  more  practical  mat- 
ters, corresponding  as  it  were  to  what  I  have  said  before  about 
the  inexactness  of  geometrical  intuition.  I  believe  that  the 
more  or  less  close  relation  of  any  applied  science  to  mathematics 
might  be  characterized  by  the  degree  of  exactness  attained, 
or  attainable,  in  its  numerical  results.  Indeed,  a  rough  classifi- 
cation of  these  sciences  could  be  based  simply  on  the  number 
of  significant  figures  averaged  in  each.  Astronomy  (and  some 
branches  of  physics)  would  here  take  the  first  rank ;  the  num- 


MATHEMATICS   AND   THE   APPLIED   SCIENCES.  47 

her  of  significant  figures  attained  may  here  be  placed  as  high  as 
seven,  and  functions  higher  than  the  elementary  transcendental 
functions  can  be  used  to  advantage.  Chemistry  would  probably 
be  found  at  the  other  end  of  the  scale,  since  in  this  science 
rarely  more  than  two  or  three  significant  figures  can  be  relied 
upon.  Geometrical  drawing,  with  perhaps  3  to  4  figures,  would 
rank  between  these  extremes ;  and  so  we  might  go  on. 

The  ordinary  mathematical  treatment  of  any  applied  science 
substitutes  exact  axioms  for  the  approximate  results  of  experi- 
ence, and  deduces  from  these  axioms  the  rigid  mathematical 
conclusions.  In  applying  this  method  it  must  not  be  forgotten 
that  mathematical  developments  transcending  the  limit  of  exact- 
ness of  the  science  are  of  no  practical  value.  It  follows  that  a 
large  portion  of  abstract  mathematics  remains  without  finding 
any  practical  application,  the  amount  of  mathematics  that  can 
be  usefully  employed  in  any  science  being  in  proportion  to  the 
degree  of  accuracy  attained  in  the  science.  Thus,  while  the 
astronomer  can  put  to  good  use  a  wide  range  of  mathemati- 
cal theory,  the  chemist  is  only  just  beginning  to  apply  the  first 
derivative,  i.e.  the  rate  of  change  at  which  certain  processes  are 
going  on  ;  for  second  derivatives  he  does  not  seem  to  have 
found  any  use  as  yet. 

As  examples  of  extensive  mathematical  theories  that  do  not 
exist  for  applied  science,  I  may  mention  the  distinction  between 
the  commensurable  and  incommensurable,  the  investigations  on 
the  convergency  of  Fourier's  series,  the  theory  of  non-analytical 
functions,  etc.  It  seems  to  me,  therefore,  that  Kirchhoff  makes 
a  mistake  when  he  says  in  his  Spectral-Analyse  that  absorption 
takes  place  only  when  there  is  exact  coincidence  between  the 
wave-lengths.  I  side  with  Stokes,  who  says  that  absorption 
takes  place  in  the  vicinity  of  such  coincidence.  Similarly,  when 
the  astronomer  says  that  the  periods  of  two  planets  must  be 
exactly  commensurable  to  admit  the  possibility  of  a  collision, 


48  LECTURE   VI. 

this  holds  only  abstractly,  for  their  mathematical  centres  ;  and  it 
must  be  remembered  that  such  things  as  the  period,  the  mass, 
etc.,  of  a  planet  cannot  be  exactly  denned,  and  are  changing  all 
the  time.  Indeed,  we  have  no  way  of  ascertaining  whether 
two  astronomical  magnitudes  are  incommensurable  or  not ;  we 
can  only  inquire  whether  their  ratio  can  be  expressed  approxi- 
mately by  two  small  integers.  The  statement  sometimes  made 
that  there  exist  only  analytic  functions  in  nature  is  in  my 
opinion  absurd.  All  we  can  say  is  that  we  restrict  ourselves 
to  analytic,  and  even  only  to  simple  analytic,  functions  because 
they  afford  a  sufficient  degree  of  approximation.  Indeed,  we 
have  the  theorem  (of  Weierstrass)  that  any  continuous  function 
can  be  approximated  to,  with  any  required  degree  of  accuracy, 
by  an  analytic  function.  Thus  if  $(x)  be  our  continuous  func- 
tion, and  &  a  small  quantity  representing  the  given  limit  of 
exactness  (the  width  of  the  strip  that  we  substitute  for  the 
curve),  it  is  always  possible  to  determine  an  analytic  function 
f(x)  such  that 

<£(*)  =/(*)  +e,  where  |  e  |  <  |  8  |, 

within  the  given  limits. 

All  this  suggests  the  question  whether  it  would  not  be  pos- 
sible to  create  a,  let  us  say,  abridged  system  of  mathematics 
adapted  to  the  needs  of  the  applied  sciences,  without  passing 
through  the  whole  realm  of  abstract  mathematics.  Such  a 
system  would  have  to  include,  for  example,  the  researches  of 
Gauss  on  the  accuracy  of  astronomical  calculations,  or  the  more 
recent  and  highly  interesting  investigations  of  Tchebycheff  on 
interpolation.  The  problem,  while  perhaps  not  impossible,  seems 
difficult  of  solution,  mainly  on  account  of  the  somewhat  vague 
and  indefinite  character  of  the  questions  arising. 

I  hope  that  what  I  have  here  said  concerning  the  use  of 
mathematics  in  the  applied  sciences  will  not  be  interpreted 


MATHEMATICS   AND   THE   APPLIED   SCIENCES. 


49 


as  in  any  way  prejudicial  to  the  cultivation  of  abstract  mathe- 
matics as  a  pure  science.  Apart  from  the  fact  that  pure 
mathematics  cannot  be  supplanted  by  anything  else  as  a  means 
for  developing  the  purely  logical  powers  of  the  mind,  there 
must  be  considered  here  as  elsewhere  the  necessity  of  the 
presence  of  a  few  individuals  in  each  country  developed  in  a 
far  higher  degree  than  the  rest,  for  the  purpose  of  keeping 
up  and  gradually  raising  the  general  standard.  Even  a  slight 
raising  of  the  general  level  can  be  accomplished  only  when 
some  few  minds  have  progressed  far  ahead  of  the  average. 

Moreover,  the  "abridged"  system  of  mathematics  referred 
to  above  is  not  yet  in  existence,  and  we  must  for  the  present 
deal  with  the  material  at  hand  and  try  to  make  the  best  of  it. 

Now,  just  here  a  practical  difficulty  presents  itself  in  the 
teaching  of  mathematics,  let  us  say  of  the  elements  of  the 
differential  and  integral  calculus.  The  teacher  is  confronted 
with  the  problem  of  harmonizing  two  opposite  and  almost  con- 
tradictory requirements.  On  the  one  hand,  he  has  to  consider 
the  limited  and  as  yet  undeveloped  intellectual  grasp  of  his 
students  and  the  fact  that  most  of  them  study  mathematics 
mainly  with  a  view  to  the  practical  applications  ;  on  the  other, 
his  conscientiousness  as  a  teacher  and  man  of  science  would 
seem  to  compel  him  to  detract  in  nowise  from  perfect  mathe- 
matical rigour  and  therefore  to  introduce  from  the  beginning 
all  the  refinements  and  niceties  of  modern  abstract  mathe- 
matics. In  recent  years  the  university  instruction,  at  least  in 
Europe,  has  been  tending  more  and  more  in  the  latter  direc- 
tion ;  and  the  same  tendencies  will  necessarily  manifest  them- 
selves in  this  country  in  the  course  of  time.  The  second 
edition  of  the  Cours  d 'analyse  of  Camille  Jordan  may  be 
regarded  as  an  example  of  this  extreme  refinement  in  laying 
the  foundations  of  the  infinitesimal  calculus.  To  place  a  work 
of  this  character  in  the  hands  of  a  beginner  must  necessarily 


50  LECTURE   VI. 

have  the  effect  that  at  the  beginning  a  large  part  of  the  sub- 
ject will  remain  unintelligible,  and  that,  at  a  later  stage,  the 
student  will  not  have  gained  the  power  of  making  use  of 
the  principles  in  the  simple  cases  occurring  in  the  applied 
sciences. 

It  is  my  opinion  that  in  teaching  it  is  not  only  admissible, 
but 'absolutely  necessary,  to  be  less  abstract  at  the  start,  to 
have  constant  regard  to  the  applications,  and  to  refer  to  the 
refinements  only  gradually  as  the  student  becomes  able  to 
understand  them.  This  is,  of  course,  nothing  but  a  universal 
pedagogical  principle  to  be  observed  in  all  mathematical 
instruction. 

Among  recent  German  works  I  may  recommend  for  the  use 
of  beginners,  for  instance,  Kiepert's  new  and  revised  edition  of 
Stegemann's  text-book ;  *  this  work  seems  to  combine  sim- 
plicity and  clearness  with  sufficient  mathematical  rigour.  On 
the  other  hand,  it  is  a  matter  of  course  that  for  more  advanced 
students,  especially  for  professional  mathematicians,  the  study 
of  works  like  that  of  Jordan  is  quite  indispensable. 

I  am  led  to  these  remarks  by  the  consciousness  of  a  growing 
danger  in  the  higher  educational  system  of  Germany,  —  the 
danger  of  a  separation  between  abstract  mathematical  science 
and  its  scientific  and  technical  applications.  Such  separation 
could  only  be  deplored ;  for  it  would  necessarily  be  followed  by 
shallowness  on  the  side  of  the  applied  sciences,  and  by  isolation 
on  the  part  of  pure  mathematics. 

*  Grundriss  der  Differential-  und  Integral- Rechnung,  6te  Auflage,  herausgegeben 
von  Kiepert,  Hannover,  Helvving,  1892. 


LECTURE  VII.  :   THE   TRANSCENDENCY   OF   THE 
NUMBERS  e  AND    TT. 

(September  4,  1893.) 

LAST  Saturday  we  discussed  inexact  mathematics ;  to-day  we 
shall  speak  of  the  most  exact  branch  of  mathematical  science. 

It  has  been  sb^pwn  by  G.  Cantor  that  there  are  two  kinds 
of  infinite  manifolcraegses :  (a)  countable  (abzalilbare)  manifold- 


nesses,  whose  quantities^Q&x  be  numbered  or  enumerated  so  that 
to  each  quantity  a  definite  pl^je  can  be  assigned  in  the  system  ; 

and  (b)  non-countable  manifoldness'&Sj^for  which  this  is  not  possi- 

^^^KJ 
ble.    To  the  former  group  belong  not  oo^uthe  rational  numbers, 

but  also  the  so-called  algebraic  numbers,  z.£<Sall  numbers  defined 
by  an  algebraic  equation, 

a  +  a^x  +  a.^x?  +  •  •  •  +  anxn  =  o 

with  integral  coefficients  (n  being  of  course  a  positive  integer). 
As  an  example  of  a  non-countable  manifoldness  I  may  mention 
the  totality  of  all  numbers  contained  in  a  continuum.,  such  as 
that  formed  by  the  points  of  the  segment  of  a  straight  line. 
Such  a  continuum  contains  not  only  the  rational  and  algebraic 
numbers,  but  also  the  so-called  transcendental  numbers.  The 
actual  existence  of  transcendental  numbers  which  thus  naturally 
follows  from  Cantor's  theory  of  manifoldnesses  had  been  proved 
before,  from  considerations  of  a  different  order,  by  Liouville. 
With  this,  however,  is  not  yet  given  any  means  for  deciding 
whether  any  particular  number  is  transcendental  or  not.  But 

5' 


52  LECTURE   VII. 

during  the  last  twenty  years  it  has  been  established  that  the 
two  fundamental  numbers  e  and  TT  are  really  transcendental. 
It  is  my  object  to-day  to  give  you  a  clear  idea  of  the  very 
simple  proof  recently  given  by  Hilbert  for  the  transcendency  of 
these  two  numbers. 

The  history  of  this  problem  is  short.  Twenty  years  ago, 
Hermite*  first  established  the  transcendency  of  e\  i.e.  he 
showed,  by^somewhat  complicated  methods,  that  the  number  e 
cannot  be  the  root  of  an  algebraic  equation  with  integral 
coefficients.  Nine  years  later,  Lindemann,f  taking  the  develop- 
ments of  Hermite  as  his  point  of  departure,  succeeded  in 
proving  the  transcendency  of  TT.  Lindemann's  work  was 
verified  soon  after  by  Weierstrass. 

The  proof  that  TT  is  a  transcendental  number  will  forever 
mark  an  epoch  in  mathematical  science.  It  gives  the  final 
answer  to  the  problem  of  squaring  the  circle  and  settles  this 
vexed  question  once  for  all.  This  problem  requires  to  derive 
the  number  TT  by  a  finite  number  of  elementary  geometrical 
processes,  i.e.  with  the  use  of  the  ruler  and  compasses  alone. 
As  a  straight  line  and  a  circle,  or  two  circles,  have  only  two 
intersections,  these  processes,  or  any  finite  combination  of 
them,  can  be  expressed  algebraically  in  a  comparatively  simple 
form,  so  that  a  solution  of  the  problem  of  squaring  the  circle 
would  mean  that  TT  can  be  expressed  as  the  root  of  an  algebraic 
equation  of  a  comparatively  simple  kind,  viz.  one  that  is  solvable 
by  square  roots.  Lindemann's  proof  shows  that  TT  is  not  the 
root  of  any  algebraic  equation. 

The  proof  of  the  transcendency  of  TT  will  hardly  diminish  the 
number  of  circle-squarers,  however ;  for  this  class  of  people  has 
always  shown  an  absolute  distrust  of  mathematicians  and  a 


*  Comptes  rendus,  Vol.  77  (1873),  p.  18,  etc. 
t  Math.  Annalen,  Vol.  2O  (1882),  p.  213. 


TRANSCENDENCY  OF  THE  NUMBERS  e  AND  IT. 


53 


contempt  for  mathematics  that  cannot  be  overcome  by  any 
amount  of  demonstration.  But  Hilbert's  simple  proof  will 
surely  be  appreciated  by  all  those  who  take  interest  in  the 
establishment  of  mathematical  truths  of  fundamental  impor- 
tance. This  demonstration,  which  includes  the  case  of  the 
number  e  as  well  as  that  of  TT,  was  published  quite  recently 
in  the  Gottinger  Nachrichten*  Immediately  after  f  Hurwitz 
published  a  proof  for  the  transcendency  of  e  based  on  still 
more  elementary  principles  ;  and  finally,  Gordan  \  gave  a  fur- 
ther simplification.  All  three  of  these  papers  will  be  reprinted 
in  the  next  Heft  of  the  Math.  Annalen.§  The  problem  has 
thus  been  reduced  to  such  simple  terms  that  the  proofs  for 
the  transcendency  of  e  and  TT  should  henceforth  be  introduced 
into  university  teaching  everywhere. 

Hilbert's  demonstration  is  based  on  two  propositions.  One 
of  these  simply  asserts  the  transcendency  of  et  i.e.  the  impos- 
sibility of  an  equation  of  the  form 

a  +  a^e  -\-  a*/  -\.  -----  \-anen  =  o,  (i) 

where  a,  av  a^  ...  an  are  integral  numbers.  This  is  the  original 
proposition  of  Hermite.  To  prove  the  transcendency  of  TT, 
another  proposition  (originally  due  to  Lindemann)  is  required, 
which  asserts  the  impossibility  of  an  equation  of  the  form 

a  _|_  es,  +  eh  _|  -----  (-^-.  =  0,  (2) 

where  a  is  an  integer,  and  the  exponents  are  algebraic  numbers, 
viz.  the  roots  of  an  algebraic  equation 


b,  &v  b^,  ...  bm  being  integers. 


*  1893,  No.  2,  p.  113.  J  Comptes  rendus,  1893,  P-  IO4°- 

t  /£.,  No.  4.  §  Vol.  43  (1894),  pp.  216-224. 


54  LECTURE   VII. 

It  will  be  noticed  that  the  latter  proposition  really  includes 
the  former  as  a  special  case ;  for  it  is  of  course  possible  that 
the  /3's  are  rational  integral  numbers,  and  whenever  some  of  the 
roots  of  the  equation  for  /3  are  equal,  the  corresponding  terms 
in  the  equation  (2)  will  combine  into  a  single  term  of  the  form 
a^.  The  former  proposition  is  therefore  introduced  only  for 
the  sake  of  simplicity. 

The  central  idea  of  the  proof  of  the  impossibility  of  equation 
(i)  consists  in  introducing  for  the  quantities  i  \e  -.e2: ...  :en,  in 
which  the  equation  is  homogeneous,  proportional  quantities 

/0  4-  *o :  /i  +  *i  =  h  +  *2  :•••:/„  4-  *„, 

selected  so  that  each  consists  of  an  integer  /  and  a  very  small 
fraction  e.  The  equation  then  assumes  the  form 

(a/o  4-  0i/i  H \-  0n/»)  4-  (0*0  +  0i*i  -\ h0n*»)  =  o,         (3) 

and  it  can  be  shown  that  the  7's  and  e's  can  always  be  so 
selected  as  to  make  the  quantity  in  the  first  parenthesis,  which 
is  of  course  integral,  different  from  zero,  while  the  quantity  in 
the  second  parenthesis  becomes  a  proper  fraction.  Now,  as 
the  sum  of  an  integer  and  a  proper  fraction  cannot  be  equal 
to  zero,  the  equation  (i)  is  proved  to  be  impossible. 

So  much  for  the  general  idea  of  Hilbert's  proof.  It  will  be 
seen  that  the  main  difficulty  lies  in  the  proper  determination 
of  the  integers  /  and  the  fractions  e.  For  this  purpose  Hilbert 
makes  use  of  a  definite  integral  suggested  by  the  investigations 
of  Hermite,  viz.  the  integral 

/= 

where  p  is  an  integer  to  be  determined  afterwards.  Multiply- 
ing equation  (i)  term  for  term  by  this  integral  and  dividing 
by  p\,  this  equation  can  evidently  be  put  into  the  form 


TRANSCENDENCY   OF  THE  NUMBERS  e  AND  TT. 


55 


- 

£  ^ 


or  designating  for  shortness  the  quantities  in  the  two  paren- 
theses by  Pl  and  Pv  respectively, 

/>  +  />  =  o. 

Now  it  can  be  proved  that  the  coefficients  of  #,  d^,  a2,"-an 
in  /^  are  all  integers,  that  p  can  be  so  selected  as  to  make 
Pl  different  from  zero,  and  that  at  the  same  time  p  can  be 
taken  so  large  as  to  make  P2  as  small  as  we  please.  Thus, 
equation  (i)  will  be  reduced  to  the  impossible  form  (3). 

We  proceed  to  prove  these  properties  of  Pl  and  Pz.  The 
integral  J  is  readily  seen  to  be  an  integer  divisible  by  p!, 
owing  to  the  well-known  relation  \  ^e~*dz-=p\.  Similarly, 
by  substituting  z  =  s'+i,  z  =  z'  +  2,  •••2=2'  +  n,  it  can  be  shown 
that  e  (  ,<?(  ,...en\  are  integers  divisible  by  (p+i)\.  It 
follows  that  P1  is  an  integer,  viz. 


If,  therefore,  p  be  selected  so  as  to  make  the  right-hand  mem- 
ber of  this  congruence  not  divisible  by  p-f-i,  the  whole  expres- 
sion Pl  is  different  from  zero. 

As  regards  the  condition  that  Pz  should  be  made  as  small 
as  we  please,  it  can  evidently  be  fulfilled  by  selecting  a  suffi- 
ciently large  value  for  p  ;  this  is  of  course  consistent  with 
the  condition  of  making  J  not  divisible  by  p+i.  For  by  the 
theorem  of  mean  values  (Mittelwertsatz)  the  integrals  can  be 
replaced  by  powers  of  constant  quantities  with  p  in  the  expo- 


LECTURE   VII. 


nent  ;  and  the  rate  of  increase  of  a  power  is,  for  sufficiently 
large  values  of  p,  always  smaller  than  that  of  the  factorial  which 
occurs  in  the  denominator. 

The  proof  of  the  impossibility  of  equation  (2)  proceeds  on 
precisely  analogous  lines.  Instead  of  the  integral  J  we  have 
now  to  use  the  integral 


-  A)  («  -  A)  •••( 
the  #'s  being  the  roots  of  the  algebraic  equation 


This  integral  is  decomposed  as  follows  : 

/»00  /»fl  /»00 

f  -JC+JT- 

»/0  «/0  »/|8 

where   of   course   the   path   of  integration    must   be   properly 

determined  for  complex  values  of  ft.     For  the  details  I  must 

refer  you  to  Hilbert's  paper. 

Assuming  the  impossibility  of  equation  (2),  the  transcendency 

of  TT  follows  easily  from  the  following  considerations,  originally 

given  by  Lindemann.  We  notice 
first,  as  a  consequence  of  our  the- 
orem, that,  with  the  exception  of 
the  point  x=o,  y  =  I,  the  exponen- 
tial curve  y=e*  has  no  algebraic 

x_      point,  i.e.  no  point  both  of  whose 

co-ordinates  are  algebraic  num- 
bers. In  other  words,  however 
densely  the  plane  may  be  covered 


Fig.  12. 


with  algebraic  points,  the  exponential  curve  (Fig.  12)  manages 
to  pass  along  the  plane  without  meeting  them,  the  single  point 
(o,  i)  excepted.  This  curious  result  can  be  deduced  as  follows 
from  the  impossibility  of  equation  (2).  Let^/  be  any  algebraic 


TRANSCENDENCY  OF  THE  NUMBERS  e  AND  TT. 


57 


quantity,  i.e.  a  root  of  any  algebraic  equation,  and  let  yv  j2,  ••• 
be  the  other  roots  of  the  same  equation  ;  let  a  similar  notation 
be  used  for  x.  Then,  if  the  exponential  curve  have  any  alge- 
braic point  (x,  y),  (besides  x=o,  y=  i),  the  equation 


'i-<*')  0-2  - 


must  evidently  be  fulfilled.  But  this  equation,  when  multiplied 
out,  has  the  form  of  equation  (2),  which  has  been  shown  to  be 
impossible. 

As  second  step  we  have  only  to  apply  the  well-known  identity 


i  = 


which  is  a  special  case  of  y=e*.     Since  in  this  identity  y=  i  is 
algebraic,  x=iir  must  be  transcendental. 


LECTURE  VIII. :    IDEAL   NUMBERS. 

(September  5,  1893.) 

THE  theory  of  numbers  is  commonly  regarded  as  something 
exceedingly  difficult  and  abstruse,  and  as  having  hardly  any 
connection  with  the  other  branches  of  mathematical  science. 
This  view  is  no  doubt  due  largely  to  the  method  of  treatment 
adopted  in  such  works  as  those  of  Kummer,  Kronecker,  Dede- 
kind,  and  others  who  have,  in  the  past,  most  contributed  to  the 
advancement  of  this  science.  Thus  Kummer  is  reported  as 
having  spoken  of  the  theory  of  numbers  as  the  only  pure 
branch  of  mathematics  not  yet  sullied  by  contact  with  the 
applications. 

Recent  investigations,  however,  have  made  it  clear  that  there 
exists  a  very  intimate  correlation  between  the  theory  of  num- 
bers and  other  departments  of  mathematics,  not  excluding 
geometry. 

As  an  example  I  may  mention  the  theory  of  the  reduction 
of  binary  quadratic  forms  as  treated  in  the  Elliptische  Modul- 
functionen.  An  extension  of  this  method  to  higher  dimensions 
is  possible  without  serious  difficulties.  Another  example  you 
will  remember  from  the  paper  by  Minkowski,  Ueber  Eigen- 
schaften  von  ganzen  Zahlen,  die  dnrch  rdumliche  Anschauung 
erschlossen  sind,  which  I  had  the  pleasure  of  presenting  to 
you  in  abstract  at  the  Congress  of  Mathematics.  Here  geom- 
etry is  used  directly  for  the  development  of  new  arithmetical 

ideas. 

58 


IDEAL   NUMBERS.  59 

To-day  I  wish  to  speak  on  the  composition  of  binary  algebraic 
forms,  a  subject  first  discussed  by  Gauss  in  his  Disquisitiones 
arithmetics*  and  of  Kummer's  corresponding  theory  of  ideal 
numbers.  Both  these  subjects  have  always  been  considered  as 
very  abstruse,  although  Dirichlet  has  somewhat  simplified  the 
treatment  of  Gauss.  I  trust  you  will  find  that  the  geometrical 
considerations  by  means  of  which  I  shall  treat  these  questions 
introduce  so  high  a  degree  of  simplicity  and  clearness  that  for 
those  not  familiar  with  the  older  treatment  it  must  be  difficult 
to  realize  why  the  subject  should  ever  have  been  regarded  as 
so  very  intricate.  These  considerations  were  indicated  by 
myself  in  the  Gottinger  Nachrichten  for  January,  1893  ;  and 
at  the  beginning  of  the  summer  semester  of  the  present  year 
I  treated  them  in  more  extended  form  in  a  course  of  lectures.  I 
have  since  learned  that  similar  ideas  were  proposed  by  Poincare 
in  1 88 1  ;  but  I  have  not  yet  had  sufficient  leisure  to  make  a 
comparison  of  his  work  with  my  own. 

I  write  a  binary  quadratic  form  as  follows  : 

/=  ax2  +  bxy  +  cy*, 

i.e.  without  the  factor  2  in  the  second  term ;  some  advantages 
of  this  notation  were  recently  pointed  out  by  H.  Weber,  in 
the  Gottinger  Nachrichten,  1892-93.  The  quantities  a,  b,  c,  x, 
y  are  here  of  course  all  assumed  to  be  integers. 

It  is  to  be  noticed  that  in  the  theory  of  numbers  a  common 
factor  of  the  coefficients  a,  b,  c  cannot  be  introduced  or  omitted 
arbitrarily,  as  in  projective  geometry ;  in  other  words,  we  are 
concerned  with  the  form,  not  with  an  equation.  Hence  we 
make  the  supposition  that  the  coefficients  a,  bt  c  have  no 
common  factor ;  a  form  of  this  character  is  called  a  primitive 
form. 

*  In  the  5th  section  ;  see  Gauss's  Wcrkc,  Vol.  I,  p.  239. 


60  LECTURE   VIII. 

As  regards  the  discriminant 

D  =  P  -  4  af, 

we  shall  assume  that  it  has  no  quadratic  divisor  (and  hence 
cannot  be  itself  a  square),  and  that  it  is  different  from  zero. 
Thus  D  is  either  =  o  or  =  i  (mod.  4).  Of  the  two  cases, 

D  <  o  and  £>>o, 

which  have  to  be  considered  separately,  I  select  the  former  as 
being  more  simple.  Both  cases  were  treated  in  my  lectures 
referred  to  before. 

The  following  elementary  geometrical  interpretation  of  the 
binary  quadratic  form  was  given  by  Gauss,  who  was  kmuch 
inclined  to  using  geometrical  considerations  in  all  branches  of 
mathematics.  Construct  a  parallelogram  (Fig.  13)  with  two 


Fig.  13. 


adjacent  sides  equal  to  Va,  Ve,  respectively,  and  the  included 

angle  <f>  such  that  cos  <£  = — .    As  ^  —  4  ac  <  o,  a  and  c  have 

2  ^/ac 

necessarily  the  same  sign  ;  we  here  assume  that  a  and  c  are 


IDEAL   NUMBERS.  6l 

both  positive ;  the  case  when  they  are  both  negative  can 
readily  be  treated  by  changing  the  signs  throughout.  Next 
produce  the  sides  of  the  parallelogram  indefinitely,  and  draw 
parallels  so  as  to  cover  the  whole  plane  by  a  network  of 
equal  parallelograms.  I  shall  call  this  a  line-lattice  (Parallel- 
fitter). 

We  now  select  any  one  of  the  intersections,  or  vertices,  as 
origin  O,  and  denote  every  other  vertex  by  the  symbol  (x,  y), 
x  being  the  number  of  sides  V#,  y  that  of  sides  vV,  which 
must  be  traversed  in  passing  from  O  to  (x,  y).  Then  every 
value  that  the  form  f  takes  for  integral  values  of  x,  y  evidently 
represents  the  square  of  the  distance  of  the  point  (x,  y)  from 
O.  Thus  the  lattice  gives  a  complete  geometrical  representa- 
tion of  the  binary  quadratic  form.  The  discriminant  D  has 
also  a  simple  geometrical  interpretation,  the  area  of  each  paral- 
lelogram being  =1V—  D. 

Now,  in  the  theory  of  numbers,  two  forms 

f=ax*  +  bxy  +  fyi    and    /'  =  a '*'2  +  b '•*'/  +  ''/* 

are  regarded  as  equivalent  if  one  can  be  derived  from  the  other 
by  a  linear  substitution  whose  determinant  is  i,  say 

x'  =  ax  +  fly,    y'  =  yx  +  Sy, 

where  «S  — $7=1,  «,  ft,  7,  B  being  integers.  All  forms  equiva- 
lent to  a  given  one  are  said  to  compose  a  class  of  quadratic 
forms ;  these  forms  have  all  the  same  discriminant.  What 
corresponds  to  this  equivalence  in  our  geometrical  representa- 
tion will  readily  appear  if  we  fix  our  attention  on  the  vertices 
only  (Fig.  14) ;  we  then  obtain  what  I  propose  to  call  a  point- 
lattice  (Punktgitter).  Such  a  network  of  points  can  be  con- 
nected in  various  ways  by  two  sets  of  parallel  lines ;  i.e.  the 
point-lattice  represents  an  infinite  number  of  line-lattices.  Now 
it  results  from  an  elementary  investigation  that  the  point- 


62  LECTURE   VIII. 

lattice  is  the  geometrical  image  of  the  class  of  binary  quad- 
ratic forms,  the  infinite  number  of  line-lattices  contained  in 
the  point-lattice  corresponding  exactly  to  the  infinite  number 
of  binary  forms  contained  in  the  class. 


Fig.  14. 

It  is  further  known  from  the  theory  of  numbers  that  to 
every  value  of  D  belongs  only  a  finite  number  of  classes  ; 
hence  to  every  D  will  correspond  a  finite  number  of  point- 
lattices,  which  we  shall  afterwards  consider  together. 

Among  the  different  classes  belonging  to  the  same  value  of 
D,  there  is  one  class  of  particular  importance,  which  I  call  the 
principal  class.  It  is  defined  as  containing  the  form 

xi-\Df 
when  D  =  o  (mod.  4),  and  the  form 


when  D  =  i  (mod.  4).  It  is  easy  to  see  that  the  correspond- 
ing lattices  are  very  simple.  When  D  =  o(mod.  4),  the  principal 
lattice  is  rectangular,  the  sides  of  the  elementary  parallelo- 


IDEAL  NUMBERS.  63 

gram  being  i  and  V  —  \D.  For /)=  i  (mod.  4),  the  parallelogram 
becomes  a  rhombus.  For  the  sake  of  simplicity,  I  shall  here 
consider  only  the  former  case. 

Let  us  now  define  complex  numbers  in  connection  with  the 
principal  lattice  of  the  rectangular  type  (Fig.  15).     The  point 


Fig.  15. 

(x,  y)  of  the  lattice  will  represent  simply  the  complex  number 


such  numbers  we  shall  call  principal  numbers. 

In  any  system  of  numbers  the  laws  of  multiplication  are  of 
prime  importance.  For  our  principal  numbers  it  is  easy  to 
prove  that  the  product  of  any  two  of  them  always  gives  a 
principal  number;  i.e.  the  system  of  principal  numbers  is,  for 
multiplication,  complete  in  itself. 

We  proceed  next  to  the  consideration  of  lattices  of  discrimi- 
nant D  that  do  not  belong  to  the  principal  class  ;  let  us  call 
them  secondary  lattices  (Nebengitter}.  Before  investigating  the 
laws  of  multiplication  of  the  corresponding  numbers,  I  must 
call  attention  to  the  fact  that  there  is  one  feature  of  arbitrari- 
ness in  our  representation  that  has  not  yet  been  taken  into 
account  ;  this  is  the  orientation  of  the  lattice,  which  may  be 
regarded  as  given  by  the  angles,  -ty  and  ^,  made  by  the  sides 


64 


LECTURE  VIII. 


Vtf,  V*:,  respectively,  with  some  fixed  initial  line  (Fig.  16). 
For  the  angle  <f>  of  the  parallelogram  we  have  evidently  </>  =  ^—  -^. 
The  point  (x,  y)  of  the  lattice  will  thus  give  the  complex  number 


-y, 


which  we  call  a  secondary  number.  The  definition  of  a  secondary 
number  is  therefore  indeterminate  as  long  as  -»/r  or  ^  is  not 
fixed. 

Now,  by  determining  ty  properly  for  every  secondary  point- 
lattice,  it  is  always  possible  to  bring  about  the  important  result 


Fig.  16. 

that  the  product  of  any  two  complex  numbers  of  all  our  lattices 
taken  together  will  again  be  a  complex  number  of  the  system, 
so  that  the  totality  of  these  complex  numbers  forms,  likewise, 
for  multiplication,  a  complete  system. 

Moreover,  the  multiplication  combines  the  lattices  in  a 
definite  way;  thus,  if  any  number  belonging  to  the  lattice  L1 
be  multiplied  into  any  number  of  the  lattice  Lv  we  always  obtain 
a  number  belonging  to  a  definite  lattice  Ls. 

These  properties  will  be  seen  to  correspond  exactly  to  the 
characteristic  properties  of  Gauss's  composition  of  algebraic 
fonns.  For  Gauss's  law  merely  asserts  that  the  product  of 


IDEAL  NUMBERS.  65 

two  ordinary  numbers  that  can  be  represented  by  two  primitive 
forms  /j,  /2  of  discriminant  D  is  always  representable  by  a 
definite  primitive  form  /3  of  discriminant  D.  This  law  is 
included  in  the  theorem  just  stated,  inasmuch  as  the  values  of 
V/i,  V/Cj,  V^  represent  the  distances  of  the  points  in  the 
lattices  from  the  origin.  At  the  same  time  we  notice  that 
Gauss's  law  is  not  exactly  equivalent  to  our  theorem,  since 
in  the  multiplication  of  our  complex  numbers,  not  only  the 
distances  are  multiplied,  but  the  angles  <f>  are  added. 

It  is  not  impossible  that  Gauss  himself  made  use  of  similar 
considerations  in  deducing  his  law,  which,  taken  apart  from  this 
geometrical  illustration,  bears  such  an  abstruse  character. 

It  now  remains  to  explain  what  relation  these  investigations 
have  to  the  ideal  numbers  of  Kummer.  This  involves  the 
question  as  to  the  division  of  our  complex  numbers  and  their 
resolution  into  primes. 

In  the  ordinary  theory  of  real  numbers,  every  number  can 
be  resolved  into  primes  in  only  one  way.  Does  this  fundamental 
law  hold  for  our  complex  numbers  ?  In  answering  this  question 
we  must  distinguish  between  the  system  formed  by  the  totality 
of  all  our  complex  numbers  and  the  system  of  principal  numbers 
alone.  For  the  former  system  the  answer  is  :  yes,  every  com- 
plex number  can  be  decomposed  into  complex  primes  in  only 
one  way.  We  shall  not  stop  to  consider  the  proof  which  is 
directly  contained  in  the  ordinary  theory  of  binary  quadratic 
forms.  But  if  we  proceed  to  the  consideration  of  the  system 
of  principal  numbers  alone,  the  matter  is  different.  There 
are  cases  when  a  principal  number  can  be  decomposed  in 
more  than  one  way  into  prime  factors,  i.e.  principal  numbers 
not  decomposable  into  principal  factors.  Thus  it  may  happen 
that  we  have  mlm2  =  nln2;  mv  m2,  nv  ;/2  being  principal  primes. 
The  reason  is,  that  these  principal  numbers  are  no  longer  primes 


66  LECTURE   VIII. 

if  we  adjoin  the  secondary  numbers,  but  are  decomposable  as 
follows  : 


a>  @,  7,  8  being  primes  in  the  enlarged  system.  In  investigating 
the  laws  of  division  it  is  therefore  not  convenient  to  consider  the 
principal  system,  by  itself  ;  it  is  best  to  introduce  the  secondary 
systems.  Kummer,  in  studying  these  questions,  had  originally 
at  his  disposal  only  the  principal  system  ;  and  noticing  the 
imperfection  of  the  resulting  laws  of  division,  he  introduced 
by  definition  his  ideal  numbers  so  as  to  re-establish  the  ordinary 
laws  of  division.  These  ideal  numbers  of  Kummer  are  thus 
seen  to  be  nothing  but  abstract  representatives  of  our  secondary 
numbers.  The  whole  difficulty  encountered  by  every  one  when 
first  attacking  the  study  of  Kummer's  ideal  numbers  is  there- 
fore merely  a  result  of  his  mode  of  presentation.  By  introduc- 
ing from  the  beginning  the  secondary  numbers  by  the  side  of 
the  principal  numbers,  no  difficulty  arises  at  all. 

It  is  true  that  we  have  here  spoken  only  of  complex  numbers 
containing  square  roots,  while  the  researches  of  Kummer  him- 
self and  of  his  followers,  Kronecker  and  Dedekind,  embrace  all 
possible  algebraic  numbers.  But  our  methods  are  of  universal 
application  ;  it  is  only  necessary  to  construct  lattices  in  spaces 
of  higher  dimensions.  It  would  carry  us  too  far  to  enter  into 
details. 


LECTURE  IX.:    THE   SOLUTION   OF   HIGHER   ALGE- 
BRAIC  EQUATIONS. 

(September  6,  1893.) 

FORMERLY  the  "  solution  of  an  algebraic  equation  "  used  to 
mean  its  solution  by  radicals.  All  equations  whose  solutions 
cannot  be  expressed  by  radicals  were  classed  simply  as  insoluble, 
although  it  is  well  known  that  the  Galois  groups  belonging  to 
such  equations  may  be  very  different  in  character.  Even  at 
the  present  time  such  ideas  are  still  sometimes  found  prevail- 
ing; and  yet,  ever  since  the  year  1858,  a  very  different  point  of 
view  should  have  been  adopted.  This  is  the  year  in  which 
Hermite  and  Kronecker,  together  with  Brioschi,  found  the 
solution  of  the  equation  of  the  fifth  degree,  at  least  in  its 
fundamental  ideas. 

This  solution  of  the  quintic  equation  is  often  referred  to  as 
a  "solution  by  elliptic  functions";  but  this  expression  is  not 
accurate,  at  least  not  as  a  counterpart  to  the  "solution  by 
radicals."  Indeed,  the  elliptic  functions  enter  into  the  solution 
of  the  equation  of  the  fifth  degree,  as  logarithms  might  be  said 
to  enter  into  the  solution  of  an  equation  by  radicals,  because 
the  radicals  can  be  computed  by  means  of  logarithms.  The 
solution  of  an  equation  will,  in  the  present  lecture,  be  regarded 
as  consisting  in  its  reduction  to  certain  algebraic  normal  equa- 
tions. That  the  irrationalities  involved  in  the  latter  can,  in 
the  case  of  the  quintic  equation,  be  computed  by  means  of 
tables  of  elliptic  functions  (provided  that  the  proper  tables  of 

67 


68  LECTURE   IX. 

the  corresponding  class  of  elliptic  functions  were  available) 
is  an  additional  point  interesting  enough  in  itself,  but  not  to 
be  considered  by  us  to-day. 

I  have  simplified  the  solution  of  the  quintic,  and  think  that 
I  have  reduced  it  to  the  simplest  form,  by  introducing  the 
icosahedron  equation  as  the  proper  normal  equation.*  In  other 
words,  the  icosahedron  equation  determines  the  typical  irra- 
tionality to  which  the  solution  of  the  equation  of  the  fifth 
degree  can  be  reduced.  This  method  is  capabje  of  being  so 
generalized  as  to  embrace  a  whole  theory  of  the  solution  of 
higher  algebraic  equations ;  and  to  this  I  wish  to  devote  the 
present  lecture. 

It  may  be  well  to  state  that  I  speak  here  of  equations  with 
coefficients  that  are  not  fixed  numerically ;  the  equations  are 
considered  from  the  point  of  view  of  the  theory  of  functions, 
the  coefficients  corresponding  to  the  independent  variables. 

In  saying  that  an  equation  is  solvable  by  radicals  we  mean 
that  it  is  reducible  by  algebraic  processes  to  so-called  pure 

equations, 

rin  =  z, 

where  z  is  a  known  quantity ;  then  only  the  new  question 
arises,  how  i^=-\Tz  can  be  computed.  Let  us  compare  from 
this  point  of  view  the  icosahedron  equation  with  the  pure 
equation. 

The  icosahedron  equation  is  the  following  equation  of  the 
6oth  degree : 

^30?)     =, 

1 7287*07) 

where  H  is  a  numerical  expression  of  the  2Oth,  f  one  of  the 
1 2th  degree,  while  z  is  a  known  quantity.  For  the  actual 

*  See  my  work  Vorlesungen  tiber  das  Ikosaeder  und  die  Auflosung  der  Gleichun- 
gen  vom  fiinften  Grade,  Leipzig,  Teubner,  1884.* 


SOLUTION   OF   HIGHER  ALGEBRAIC   EQUATIONS.         69 

forms  of  H  and  f  as  well  as  other  details  I  refer  you  to  the 
Vorlesungen  iiber  das  Ikosaeder ;  I  wish  here  only  to  point 
out  the  characteristic  properties  of  this  equation. 

(i)  Let  77  be  any  one  of  the  roots ;  then  the  60  roots  can 
all  be  expressed  as  linear  functions  of  77,  with  known  coeffi- 
cients, such  as  for  instance, 


•  ** 


2<ir 


where  e=e&.    'These  60  quantities,  then,  form  a  group  of  60 
linear  substitutions. 


2=-oo 


Fig.  17. 


(2)  Let  us  next  illustrate  geometrically  the  dependence  of  17 
on  z  by  establishing  the  conformal  representation  of  the  ^-plane 
on  the  Tj-plane,  or  rather  (by  stereographic  projection)  on  a 
sphere  (Fig.  17).  The  triangles  corre- 
sponding to  the  upper  (shaded)  half  of 
the  ^--plane  are  the  alternate  (shaded) 
triangles  on  the  sphere  determined  by 
inscribing  a  regular  icosahedron  and 
dividing  each  of  the  20  triangles  so 
obtained  into  six  equal  and  symmetrical 
triangles  by  drawing  the  altitudes  (Fig. 


Fig.  18. 


18).     This  conformal  representation  on  the  sphere  assigns  to 
every  root  a  definite  region,  and  is  therefore  equivalent  to  a 


70  LECTURE   IX. 

perfect  separation  of  the  60  roots.  On  the  other  hand,  it  cor- 
responds in  its  regular  shape  to  the  60  linear  substitutions 
indicated  above. 

(3)  If,  by  putting  rn=yi/yv  we  make  the  60  expressions 
of  the  roots  homogeneous,  the  different  values  of  the  quan- 
tities y  will  all  be  of  the  form 


<*y\ 


and    therefore    satisfy    a    linear   differential    equation    of    the 
second  order 


p  and  q  being  definite  rational  functions  of  z.  It  is,  of  course, 
always  possible  to  express  every  root  of  an  equation  by  means 
of  a  power  series.  In  our  case  we  reduce  the  calculation  of 
t)  to  that  of  y±  and  j/2,  and  try  to  find  series  for  these  quanti- 
ties. Since  these  series  must  satisfy  our  differential  equation 
of  the  second  order,  the  law  of  the  series  is  comparatively 
simple,  any  term  being  expressible  by  means  of  the  two 
preceding  terms. 

(4)  Finally,  as  mentioned  before,  the  calculation  of  the 
roots  may  be  abbreviated  by  the  use  of  elliptic  functions, 
provided  tables  of  such  elliptic  functions  be  computed  before- 
hand. 

Let  us  now  see  what  corresponds  to  each  of  these  four 
points  in  the  case  of  the  pure  equation  tf=s.  The  results  are 
well  known  : 

(i)  All  the  «  roots  can  be  expressed  as  linear  functions 
of  any  one  of  them,  77  : 


e  being  a  primitive  ;/th  root  of  unity. 


SOLUTION   OF   HIGHER   ALGEBRAIC   EQUATIONS.         71 

(2)  The  conformal  representation  (Fig.  19)  gives  the  division 
of  the  sphere  into  2  n  equal  lunes  whose  great  circles  all  pass 
through  the  same  two  points. 


Fig.  19. 


(3)    There  is  a  differential  equation  of  the  first  order  in  77, 


vz., 


nz  •  t  —  r  = 


from  which  simple  series  can  be  derived  for  the  purposes  of 
actual  calculation  of  the  roots. 

(4)  If  these  series  should  be  inconvenient,  logarithms  can  be 
used  for  computation. 

The  analogy,  you  will  perceive,  is  complete.  The  principal 
difference  between  the  two  cases  lies  in  the  fact  that,  for  the 
pure  equation,  the  linear  substitutions  involve  but  one  quantity, 
while  for  the  quintic  equation  we  have  a  group  of  binary  linear 
substitutions.  The  same  distinction  finds  expression  in  the 
differential  equations,  the  one  for  the  pure  equation  being  of 
the  first  order,  while  that  for  the  quintic  is  of  the  second  order. 

Some  remarks  may  be  added  concerning  the  reduction  of  the 
general  equation  of  the  fifth  degree, 


to  the  icosahedron  equation.  This  reduction  is  possible  because 
the  Galois  group  of  our  quintic  equation  (the  square  root  of  the 
discriminant  having  been  adjoined)  is  isomorphic  with  the  group 


72  LECTURE   IX. 

of  the  60  linear  substitutions  of  the  icosahedron  equation.  This 
possibility  of  the  reduction  does  not,  of  course,  imply  an  answer 
to  the  question,  what  operations  are  needed  to  effect  the  reduc- 
tion. The  second  part  of  my  Vorlesungen  ilber  das  Ikosaeder  is 
devoted  to  the  latter  question.  It  is  found  that  the  reduction 
cannot  be  performed  rationally,  but  requires  the  introduction  of 
a  square  root.  The  irrationality  thus  introduced  is,  however,  an 
irrationality  of  a  particular  kind  (a  so-called  accessory  irration- 
ality) ;  for  it  must  be  such  as  not  to  reduce  the  Galois  group  of 
the  equation. 

I  proceed  now  to  consider  the  general  problem  of  an  analo- 
gous treatment  of  higher  equations  as  first  given  by  me  in  the 
Math.  Annalen,  Vol.  15  (1879).*  I  must  remark,  first  of  all, 
that  for  an  accurate  exposition  it  would  be  necessary  to  dis- 
tinguish throughout  between  the  homogeneous  and  projective 
formulations  (in  the  latter  case,  only  the  ratios  of  the  homoge- 
neous variables  are  considered).  Here  it  may  be  allowed  to 
disregard  this  distinction. 

Let  us  consider  the  very  general  problem  :  a  finite  group  of 
homogeneous  linear  substitutions  of  n  variables  being  given,  to 
calculate  the  values  of  the  n  variables  from  the  invariants  of  the 
group. 

This  problem  evidently  contains  the  problem  of  solving  an 
algebraic  equation  of  any  Galois  group.  For  in  this  case  all 
rational  functions  of  the  roots  are  known  that  remain  unchanged 
by  certain  permutations  of  the  roots,  and  permutation  is,  of 
course,  a  simple  case  of  homogeneous  linear  transformation. 

Now  I  propose  a  general  formulation  for  the  treatment  of 
these  different  problems  as  follows  :  among  the  problems  having 
isomorphic  groups  we  consider  as  the  simplest  the  one  that  has  the 

*  Ueber  die  Auflosung  gewisscr  Gleichungen  vom  siebenttn  und  achttn  Grade, 
pp.  251-282. 


SOLUTION   OF   HIGHER   ALGEBRAIC   EQUATIONS.         73 

least  number  of  variables,  and  call  this  the  normal  problem.  This 
problem  must  be  considered  as  solvable  by  series  of  some  kind. 
The  question  is  to  reduce  the  other  isomorphic  problems  to  the 
normal  problem. 

This  formulation,  then,  contains  what  I  propose  as  a  gen- 
eral solution  of  algebraic  equations,  i.e.  a  reduction  of  the  equa- 
tions to  the  isomorphic  problem  with  a  minimum  number  of 
variables. 

The  reduction  of  the  equation  of  the  fifth  degree  to  the 
icosahedron  problem  is  evidently  contained  in  this  as  a  special 
case,  the  minimum  number  of  variables  being  two. 

In  conclusion  I  add  a  brief  account  showing  how  far  the  gen- 
eral problem  has  been  treated  for  equations  of  higher  degrees. 

In  the  first  place,  I  must  here  refer  to  the  discussion  by 
myself*  and  Gordan  f  of  those  equations  of  the  seventh  degree 
that  have  a  Galois  group  of  168  substitutions.  The  minimum 
number  of  variables  is  here  equal  to  three,  the  ternary  group 
being  the  same  group  of  168  linear  substitutions  that  has  since 
been  discussed  with  full  details  in  Vol.  I.  of  the  Elliptische 
Modulfunctionen.  While  I  have  confined  myself  to  an  expo- 
sition of  the  general  idea,  Gordan  has  actually  performed  the 
reduction  of  the  equation  of  the  seventh  degree  to  the  ternary 
problem.  This  is  no  doubt  a  splendid  piece  of  work ;  it  is 
only  to  be  deplored  that  Gordan  here,  as  elsewhere,  has  dis- 
dained to  give  his  leading  ideas  apart  from  the  complicated 
array  of  formulae. 

Next,  I  must  mention  a  paper  published  in  Vol.  28  (1887)  of 
the  Math.  Annalen,\  where  I  have  shown  that  for  the  general 

*  Math.  Annalen,  Vol.  15  (1879),  pp.  251-282. 

f  Ueber  Gleichungen  siebenten  Grades  mil  einer  Gruppe  von  168  Substitutionen, 
Math.  Annalen,  Vol.  20  (1882),  pp.  515-530,  and  Vol.  25  (1885),  pp.  459-521. 

\  Zur  Theorie  der  allgemeinen  Gleichungen  sechsten  und  siebenten  Grades,  pp. 
499-532. 


74  LECTURE   IX. 

equations  of  the  sixth  and  seventh  degrees  the  minimum  num- 
ber of  the  normal  problem  is  four,  and  how  the  reduction  can 
be  effected. 

Finally,  in  a  letter  addressed  to  Camille  Jordan*  I  pointed 
out  the  possibility  of  reducing  the  equation  of  the  2/th  degree, 
which  occurs  in  the  theory  of  cubic  surfaces,  to  a  normal  prob- 
lem containing  likewise  four  variables.  This  reduction  has 
ultimately  been  performed  in  a  very  simple  way  by  Burkhardt  f 
while  all  quaternary  groups  here  mentioned  have  been  con- 
sidered more  closely  by  Maschke.J 

This  is  the  whole  account  of  what  has  been  accomplished  ; 
but  it  is  clear  that  further  progress  can  be  made  on  the  same 
lines  without  serious  difficulty. 

A  first  problem  I  wish  to  propose  is  as  follows.  In  recent 
years  many  groups  of  permutations  of  6,  7,  8,  9,  ...  letters  have 
been  made  known.  The  problem  would  be  to  determine  in 
each  case  the  minimum  number  of  variables  with  which  isomor- 
phic  groups  of  linear  substitutions  can  be  formed. 

Secondly,  I  want  to  call  your  particular  attention  to  the  case 
of  the  general  equation  of  the  eighth  degree.  I  have  not  been 
able  in  this  case  to  find  a  material  simplification,  so  that  it 
would  seem  as  if  the  equation  of  the  eighth  degree  were  its 
own  normal  problem.  It  would  no.  doubt  be  interesting  to 
obtain  certainty  on  this  point. 

*  Journal  de  mathematiques,  annee  1888,  p.  169. 

f  Untersuchungen  am  dem  Gebiete  der  hyperelliptischen  Modulfunctionen.  Dritter 
Theil,  Math.  Annalen,  Vol.  41  (1893),  PP-  3!3~343- 

J  Ueber  die  quaternare,  endliche,  lineare  Substitutionsgruppe  der  Borchardfschen 
Afoduln,  Math.  Annalen,  Vol.  30  (1887),  PP-  496-515;  Aufstellung  des  vollen  For- 
mensy stems  einer  quaternaren  Gruppe  -von  51840  linear  en  Substitutionen,  ib.,  Vol. 
33  0889),  pp.  317-344;  Ueber  eine  merkwiirdige  Configuration  gerader  Linien  im 
jRaume,  ib.,  Vol.  36  (1890),  pp.  190-215. 


LECTURE    X.  :     ON    SOME    RECENT    ADVANCES     IN 
HYPERELLIPTIC   AND   ABELIAN   FUNCTIONS. 

(September  7,  1893.) 

THE  subject  of  hyperelliptic  and  Abelian  functions  is  of  such 
vast  dimensions  that  it  would  be  impossible  to  embrace  it  in 
its  whole  extent  in  one  lecture.  I  wish  to  speak  only  of  the 
mutual  correlation  that  has  been  established  between  this 
subject  on  the  one  hand,  and  the  theory  of  invariants,  projective 
geometry,  and  the  theory  of  groups,  on  the  other.  Thus  in 
particular  I  must  omit  all  mention  of  the  recent  attempts  to 
bring  arithmetic  to  bear  on  these  questions.  As  regards  the 
theory  of  invariants  and  projective  geometry,  their  introduction 
in  this  domain  must  be  considered  as  a  realization  and  farther 
extension  of  the  programme  of  Clebsch.  But  the  additional 
idea  of  groups  was  necessary  for  achieving  this  extension. 
What  I  mean  by  establishing  a  mutual  correlation  between 
these  various  branches  will  be  best  understood  if  I  explain  it 
on  the  more  familiar  example  of  the  elliptic  functions. 

To  begin  with  the  older  method,  we  have  the  fundamental 
elliptic  functions  in  the  Jacobian  form 


sin  am 


as  depending  on  two  arguments.  These  are  treated  in  many 
works,  sometimes  more  from  the  geometrical  point  of  view  of 
Riemann,  sometimes  more  from  the  analytical  standpoint  of 

75 


76  LECTURE   X. 

Weierstrass.  I  may  here  mention  the  first  edition  of  the  work 
of  Briot  and  Bouquet,  and  of  German  works  those  by  Konigs- 
berger  and  by  Thomae. 

The  impulse  for  a  new  treatment  is  due  to  Weierstrass.  He 
introduced,  as  is  well  known,  three  homogeneous  arguments, 
u,  o>v  &)2,  instead  of  the  two  Jacobian  arguments.  This  was 
a  necessary  preliminary  to  establishing  the  connection  with 
the  theory  of  linear  substitutions.  Let  us  consider  the  dis- 
continuous ternary  group  of  linear  substitutions, 

u'  =  u  -f-  0*iti>i  -f- 


where  a,  ft,  y,  &  are  integers  whose  determinant  a$—py=i, 
while  tnlt  m2  are  any  integers  whatever.  The  fundamental 
functions  of  Weierstrass's  theory, 


Ctfj, 


are  nothing  but  the  complete  system  of  invariants  of  that  group. 
It  appears,  moreover,  that  gv  gz  are  also  the  ordinary  (Cay- 
leyan)  invariants  of  the  binary  biquadratic  form  f^(x^,  xz),  on 
which  depends  the  integral  of  the  first  kind 


This  significant  feature  that  the  transcendental  invariants  turn 
out  to  be  at  the  same  time  invariants  of  the  algebraic  irration- 
ality corresponding  to  the  transcendental  theory  will  hold  in 
all  higher  cases. 

As  a  next  step  in  the  theory  of  elliptic  functions  we  have  to 
mention  the  introduction  by  Glebsch  of  the  systematic  con- 
sideration of  algebraic  curves  of  deficiency  i.  He  considered 
in  particular  the  plane  curve  of  the  third  order  (Cs)  and  the 


HYPERELLIPTIC   AND   ABELIAN    FUNCTIONS. 


77 


first  species  of  quartic  curves  (C^)  in  space,  and  showed  how 
convenient  it  is  for  the  derivation  of  numerous  geometrical 
propositions  to  regard  the  elliptic  integrals  as  taken  along  these 
curves.  The  theory  of  elliptic  functions  is  thus  broadened  by 
bringing  to  bear  upon  it  the  ideas  of  modern  projective  geometry. 

By  combining  and  generalizing  these  considerations,  I  was 
led  to  the  formulation  of  a  very  general  programme  which  may 
be  stated  as  follows  (see  Vorlesungen  iiber  die  Theorie  der  ellip- 
tischen  Modulfunctionen,  Vol.  II.). 

Beginning  with   the  discontinuous  group  mentioned  before 


u  =  u 

oV  =  awj  -f-  /2w2, 

<»2  =  ywi  +    Su)2, 

our  first  task  is  to  construct  all  its  sub-groups.  Among  these 
the  simplest  and  most  useful  are  those  that  I  have  called 
congruence  sub-groups  ;  they  are  obtained  by  putting 

mi  =  o,   m2  =  o,^\ 
«  =  i,     /3  =  o,  [-(mod.  ri). 
y  =  o,      8=1,  J 

The  second  problem  is  to  construct  the  invariants  of  all 
these  groups  and  the  relations  between  them.  Leaving  out 
of  consideration  all  sub-groups  except  these  congruence  sub- 
groups, we  have  still  attained  a  very  considerable  enlargement 
of  the  theory  of  elliptic  functions.  According  to  the  value 
assigned  to  the  number  n,  I  distinguish  different  stages  (Stufen) 
of  the  problem.  It  will  be  noticed  that  Weierstrass's  theory 
corresponds  to  the  first  stage  («=i),  while  Jacobi's  answers, 
generally  speaking,  to  the  second  («  =  2);  the  higher  stages 
have  not  been  considered  before  in  a  systematic  way. 

Thirdly,  for  the  purpose  of  geometrical  illustration,  I  apply 
Clebsch's  idea  of  the  algebraic  curve.  I  begin  by  introducing 


78  LECTURE  X. 

the  ordinary  square  root  of  the  binary  form  which  requires  the 
axis  of  x  to  be  covered  twice ;  i.e.  we  have  to  use  a  £T2  in  an 
5r  I  next  proceed  to  the  general  cubic  curve  of  the  plane 
(C3  in  an  52),  to  the  quartic  curve  in  space  of  three  dimensions 
(C±  in  an  S3),  and  generally  to  the  elliptic  curve  Cn+l  in  an  Sn. 
These  are  what  I  call  the  normal  elliptic  curves ;  they  serve  best 
to  illustrate  any  algebraic  relations  between  elliptic  functions. 

I  may  notice,  by  the  way,  that  the  treatment  here  proposed 
is  strictly  followed  in  the  Elliptische  Modulfunctionen,  except 
that  there  the  quantity  u  is  of  course  assumed  to  be  zero,  since 
this  is  precisely  what  characterizes  the  modular  functions.  I 
hope  some  time  to  be  able  to  treat  the  whole  theory  of  elliptic 
functions  (i.e.  with  u  different  from  zero)  according  to  this 
programme. 

The  successful  extension  of  this  programme  to  the  theory  of 
hyperelliptic  and  Abelian  functions  is  the  best  proof  of  its 
being  a  real  step  in  advance.  I  have  therefore  devoted  my 
efforts  for  many  years  to  this  extension ;  and  in  laying  before 
you  an  account  of  what  has  been  accomplished  in  this  rather 
special  field,  I  hope  to  attract  your  attention  to  various  lines  of 
research  along  which  new  work  can  be  spent  to  advantage. 

As  regards  the  hyperelliptic  functions,  we  may  premise  as  a 
general  definition  that  they  are  functions  of  two  variables  ult  «2, 
with  four  periods  (while  the  elliptic  functions  have  one  vari- 
able »,  and  two  periods).  Without  attempting  to  give  an 
historical  account  of  the  development  of  the  theory  of  hyper- 
elliptic functions,  I  turn  at  once  to  the  researches  that  mark 
a  progress  along  the  lines  specified  above,  beginning  with  the 
geometric  application  of  these  functions  to  surfaces  in  a  space 
of  any  number  of  dimensions. 

Here  we  have  first  the  investigation  by  Rohn  of  Kummer's 
surface,  the  well-known  surface  of  the  fourth  order,  with  16 


HYPERELLIPTIC   AND   ABELIAN   FUNCTIONS. 


79 


conical  points.  I  have  myself  given  a  report  on  this  work  in 
the  Math.  Annalen,  Vol.  27  (1886).*  If  every  mathematician  is 
struck  by  the  beauty  and  simplicity  of  the  relations  developed 
in  the  corresponding  cases  of  the  elliptic  functions  (the  CB  in 
the  plane,  etc.).  the  remarkable  configurations  inscribed  and 
circumscribed  to  the  Kummer  surface  that  have  here  been 
developed  by  Rohn  and  myself,  should  not  fail  to  elicit  interest. 

Further,  I  have  to  mention  an  extensive  memoir  by  Reichardt, 
published  in  1886,  in  the  Acta  Leopoldina,  where  the  connec- 
tion between  hyperelliptic  functions  and  Kummer's  surface  is 
summarized  in  a  convenient  and  comprehensive  form,  as  an 
introduction  to  this  branch.  The  starting-point  of  the  investi- 
gation is  taken  in  the  theory  of  line-complexes  of  the  second 
degree. 

Quite  recently  the  French  mathematicians  have  turned  their 
attention  to  the  general  question  of  the  representation  of  sur- 
faces by  means  of  hyperelliptic  functions,  and  a  long  memoir  by 
Humbert  on  this  subject  will  be  found  in  the  last  volume  of  the 
Journal  de  Mathhnatiqnes.^ 

I  turn  now  to  the  abstract  theory  of  hyperelliptic  functions. 
It  is  well  known  that  Gopel  and  Rosenhain  established  that 
theory  in  1847  in  a  manner  closely  corresponding  to  the  Jaco- 
bian  theory  of  elliptic  functions,  the  integrals 


/dx  C 

,  "2=  | 

*       J 


xdx 


taking  the  place  of  the  single  elliptic  integral  u.  Here,  then, 
the  question  arises  :  what  is  the  relation  of  the  hyperelliptic 
functions  to  the  invariants  of  the  binary  form  of  the  sixth  order 
/6(*i,  x2]  ?  In  the  investigation  of  this  question  by  myself  and 

*  Ueber  Configurationen,  welche  der  Kummer1  schen  Flache  zugleich  eingeschrieben 
und  umgesehrieben  sine/,  pp.  106-142. 

t  Theorie  generate  des  surfaces  hyperelliptiques,  annee  1893,  pp.  29-170. 


8o  LECTURE  X. 

Burkhardt,  published  in  Vol.  27  (1886)*  and  Vol.  32  (1888)  f 
of  the  Math.  Annalen,  we  found  that  the  decompositions  of 
the  form  f6  into  two  factors  of  lower  order,  /6  =  <f>^Jr5  =  ^3^3, 
had  to  be  considered.  These  being,  of  course,  irrational  decom- 
positions, the  corresponding  invariants  are  irrational  ;  and  a 
study  of  the  theory  of  such  invariants  became  necessary. 

But  another  new  step  had  to  be  taken.     The  hyperelliptic 
integrals  involve  the  form  /6  under  the  square  root, 


The  corresponding  Riemann  surface  has,  therefore,  two  leaves 
connected  at  six  points  ;  and  the  problem  arises  of  considering 
binary  forms  of  x^  x^  on  such  a  Riemann  surface,  just  as  ordi- 
narily functions  of  x  alone  are  considered  thereon.  It  can  be 
shown  that  there  exists  a  particular  kind  of  forms  called  prime- 
forms,  strictly  analogous  to  the  determinant  x-^y^—x^-^  in  the 
ordinary  complex  plane.  The  primeform  on  the  two-leaved 
Riemann  surface,  like  this  determinant  in  the  ordinary  theory, 
has  the  property  of  vanishing  only  when  the  points  (xlt  x^  and 
(y\>  y^>  co-incide  (on  the  same  leaf).  Moreover,  the  primeform 
does  not  become  infinite  anywhere.  The  analogy  to  the  deter- 
minant x^y^—x^y-^  fails  only  in  so  far  as  the  primeform  is  no 
longer  an  algebraic  but  a  transcendental  form.  Still,  all  alge- 
braic forms  on  the  surface  can  be  decomposed  into  prime 
factors.  Moreover,  these  primeforms  give  the  natural  means 
for  the  construction  of  the  ^-functions.  As  an  intermediate 
step  we  have  here  functions  called  by  me  o--functions  in  analogy 
to  the  -o--functions  of  Weierstrass's  elliptic  theory.  In  the 
papers  referred  to  (Math.  Annalen,  Vols.  27,  32)  all  these  con- 
siderations are,  of  course,  given  for  the  general  case  of  hyper- 


elliptic  functions,  the  irrationality  being  V/^+a^,  x<£),  where 
f2p+2  is  a  binary  form  of  the  order  2/4-2. 


*  Ueber  hyperelliptische  Sigmafunetionen,  pp.  431-464. 
t  PP-  35 '-380  and  381-442. 


HYPERELLIPTIC   AND   ABELIAN   FUNCTIONS.  8 1 

Having  thus  established  the  connection  between  the  ordinary 
theory  of  hyperelliptic  functions  of /  =  2  and  the  invariants  of 
the  binary  sextic,  I  undertook  the  systematic  development  of 
what  I  have  called,  in  the  case  of  elliptic  functions,  the  Stufen- 
theorie.  The  lectures  I  gave  on  this  subject  in  1887-88 
have  been  developed  very  fully  by  Burkhardt  in  the  Math. 
Annalen,  Vol.  35  (1890).* 

As  regards  the  first  stage,  which,  owing  to  the  connection 
with  the  theory  of  rational  invariants  and  covariants,  requires 
very  complicated  calculations,  the  Italian  mathematician,  Pascal, 
has  made  much  progress  (Annali  di  matematica).  In  this 
connection  I  must  refer  to  the  paper  by  Bolzaf  in  Math. 
Annalen,  Vol.  30  (1887),  where  the  question  is  discussed  in 
how  far  it  is  possible  to  represent  the  rational  invariants  of 
the  sextic  by  means  of  the  zero  values  of  the  ^-functions. 

For  higher  stages,  in  particular  stage  three,  Burkhardt  has 
given  very  valuable  developments  in  the  Math.  Annalen,  Vol. 
36  (1890),  p.  371  ;  Vol.  38  (1891),  p.  161  ;  Vol.  41  (1893),  p.  313. 
He  considers,  however,  only  the  hyperelliptic  modular  functions 
(«j  and  uz  being  assumed  to  be  zero).  The  final  aim,  which 
Burkhardt  seems  to  have  attained,  although  a  large  amount 
of  numerical  calculation  remains  to  be  filled  in,  consists  here 
in  establishing  the  so-called  multiplier-equation  for  transforma- 
tions of  the  third  order.  The  equation  is  of  the  4Oth  degree ; 
and  Burkhardt  has  given  the  general  law  for  the  formation 
of  the  coefficients. 

I  invite  you  to  compare  his  treatment  with  that  of  Krause 
in  his  book  Die  Transformation  der  hyperelliptischen  Func- 
tionen  erster  Ordnnng,  Leipzig,  Teubner,  1886.  His  investiga- 

*  Grundzilge  einer  allgemeinen  Systematik  der  hyperelliptischen  Functionen  I. 
Ordnung,  pp.  198-296. 

t  Darstellung  der  rationalen  ganzen  Invariantcn  der  Binarform  scchsten  Grades 
durch  die  Nullwerthe  der  zugehorigen  9- Functionen,  pp.  478-495. 


82  LECTURE  X. 

tions,  based  on  the  general  relations  between  ^-functions,  may 
go  farther ;  but  they  are  carried  out  from  the  purely  formal 
point  of  view,  without  reference  to  the  theories  of  invariants, 
of  groups,  or  other  allied  topics. 

So  much  as  regards  hyperelliptic  functions.  I  now  proceed 
to  report  briefly  on  the  corresponding  advances  made  in  the 
theory  of  Abelian  functions.  I  give  merely  a  list  of  papers ; 
they  may  be  classed  under  three  heads  : 

(1)  A  preliminary  question  relates  to  the  invariant  represen- 
tation of  the  integral  of  the  third  kind  on  algebraic  curves  of 
higher  deficiency.    Pick  *  has  considered  this  problem  for  plane 
curves  having  no  singular  points.     On  the  other  hand,  White, 
in  his  dissertation,!  briefly  reported  in  Math.  Annalen,  Vol.  36 
(1890),  p.  597,  and  printed  in  full  in  the  Acta  Leopoldina,  has 
treated  such  curves  in  space  as  are  the  complete  intersection 
of  two  surfaces  and  have   no   singular   point.      We  may  here 
also  notice  the  researches  of  Pick  and  Osgoodij:  on  the  so-called 
binomial  integrals. 

(2)  An  exposition  of   the  general   theory  of  forms  on  Rie- 
mann  surfaces  of  any  kind,  in  particular  a  definition  of   the 
primeform   belonging   to   each    surface,  was   given    by   myself 
in   Vol.  36    (1890)  of   the   Math.  Annalen.^     I   may  add   that 
during   the   last   year   this    subject    was    taken    up   anew   and 
farther  developed   by    Dr.    Ritter ;    see    Gbttinger  Nachrichten 
for  1893,  and  Math.  Annalen,  Vol.  44.     Dr.   Ritter  considers 
the  algebraic  forms  as  special  cases  of  more  general  forms,  the 
multiplicative  forms,  and  thus  takes  a  real  step  in  advance. 

*  Zur  Theorie  der  Abefschen  Functionen,  Math.  Annalen,  Vol.  29  (1887),  pp. 
259-271. 

t  AbeFsche  Integrate  auf  singularit'dtenfreien,  einfach  iiberdeckten,  vollstdndigen 
Schnittcurven  eines  beliebig  ausgedehnten  Raumes,  Halle,  1891,  pp.  43-128. 

J  Osgood,  Zur  Theorie  der  zum  algebraischen  Gebilde  ym  =  R(x)  gehorigen 
Abel'schen  Functionen,  Gottingen,  1890,  8vo,  61  pp. 

§  Zur  Theorie  der  AbeFschen  Functionen,  pp.  1-83. 


HYPERELLIPTIC   AND   ABELIAN   FUNCTIONS.  83 

(3)  Finally,  the  particular  case  /  =  3  has  been  studied  on  the 
basis  of  our  programme  in  various  directions.  The  normal 
curve  for  this  case  is  well  known  to  be  the  plane  quartic  C± 
whose  geometric  properties  have  been  investigated  by  Hesse 
and  others.  I  found  (Math.  Annalen,  Vol.  36)  that  these 
geometrical  results,  though  obtained  from  an  entirely  different 
point  of  view,  corresponded  exactly  to  the  needs  of  the  Abelian 
problem,  and  actually  enabled  me  to  define  clearly  the  64 
^-functions  with  the  aid  of  the  £T4.  Here,  as  elsewhere,  there 
seems  to  reign  a  certain  pre-established  harmony  in  the  develop- 
ment of  mathematics,  what  is  required  in  one  line  of  research 
being  supplied  by  another  line,  so  that  there  appears  to  be 
a  logical  necessity  in  this,  independent  of  our  individual 
disposition. 

In  this  case,  also,  I  have  introduced  <r-functions  in  the  place 
of  the  ^-functions.  The  coefficients  are  irrational  covariants 
just  as  in  the  case  /  =  2.  These  o--series  have  been  studied  at 
great  length  by  Pascal  in  the  Annali  di  Matematica.  These 
investigations  bear,  of  course,  a  close  relation  to  those  of 
Frobenius  and  Schottky,  which  only  the  lack  of  time  prevents 
me  from  quoting  in  detail. 

Finally,  the  recent  investigations  of  an  Austrian  mathemati- 
cian, Wirtinger,  must  here  be  mentioned.  First,  Wirtinger  has 
established  for  p=Z  the  analogue  to  the  Kummer  surface;  this 
is  a  manifoldness  of  three  dimensions  and  the  24th  order  in  an 
57 ;  see  Gottinger  Nachrichten  for  1889,  and  Wiener  Monatshefte, 
1890.  Though  apparently  rather  complicated,  this  manifoldness 
has  some  very  elegant  properties  ;  thus  it  is  transformed  into 
itself  by  64  collineations  and  64  reciprocations.  Next,  in 
Vol.  40  (1892),  of  the  Math.  Annalen*  Wirtinger  has  dis- 
cussed the  Abelian  functions  on  the  assumption  that  only 

*  Untersuchungen  Uber  Abetsche  Functionen  vom  Geschlechte  3,  pp.  261-312. 


84  LECTURE   X. 

rational  invariants  and  covariants  of  the  curve  of  the  fourth 
order  are  to  be  considered  ;  this  corresponds  to  the  "  first 
stage "  with  /  =  3.  The  investigation  is  full  of  new  and 
fruitful  ideas. 

In  concluding,  I  wish  to  say  that,  for  the  cases  p=2  and 
p  =  3,  while  much  still  remains  to  be  done,  the  fundamental 
difficulties  have  been  overcome.  The  great  problem  to  be 
attacked  next  is  that  of  /=4,  where  the  normal  curve  is  of  the 
sixth  order  in  space.  It  is  to  be  hoped  that  renewed  efforts 
will  result  in  overcoming  all  remaining  difficulties.  Another 
promising  problem  presents  itself  in  the  field  of  ^-functions, 
when  the  general  0-series  are  taken  as  starting-point,  and  not 
the  algebraic  curve.  An  enormous  number  of  formulae  have 
there  been  developed  by  analysts,  and  the  problem  would  be 
to  connect  these  formulae  with  clear  geometrical  conceptions 
of  the  various  algebraic  configurations.  I  emphasize  these 
special  problems  because  the  Abelian  functions  have  always 
been  regarded  as  one  of  the  most  interesting  achievements 
of  modern  mathematics,  so  that  every  advance  we  make  in 
this  theory  gives  a  standard  by  which  we  can  measure  our 
own  efficiency. 


LECTURE    XI.  :    THE    MOST    RECENT    RESEARCHES 
IN   NON-EUCLIDEAN   GEOMETRY. 

(September  8,  1893.) 

MY  remarks  to-day  will  be  confined  to  the  progress  of  non- 
Euclidean  geometry  during  the  last  few  years.  Before  report- 
ing on  these  latest  developments,  however,  I  must  briefly 
summarize  what  may  be  regarded  as  the  general  state  of 
opinion  among  mathematicians  in  this  field.  There  are  three 
points  of  view  from  which  non-Euclidean  geometry  has  been 
considered. 

(1)  First  we  have  the  point  of  view  of  elementary  geometry,  of 
which  Lobachevsky  and  Bolyai  themselves  are  representatives. 
Both  begin  with  simple  geometrical  constructions,  proceeding 
just  like  Euclid,  except  that  they  substitute  another  axiom  for 
the  axiom  of  parallels.     Thus  they  build  up  a  system  of  non- 
Euclidean  geometry  in  which  the  length  of  the  line  is  infinite, 
and  the  "measure  of  curvature"  (to  anticipate  a  term  not  used 
by  them)  is  negative.      It  is,  of  course,  possible  by  a  similar 
process  to  obtain  the  geometry  with    a   positive   measure   of 
curvature,  first  suggested  by  Riemann  ;    it   is   only  necessary 
to  formulate  the  axioms  so  as  to  make  the   length  of  a  line 
finite,  whereby  the  existence  of  parallels  is  made  impossible. 

(2)  From  the  point  of  view  of  projective  geometry,  we  begin 
by  establishing  the  system  of  projective  geometry  in  the  sense 
of   von    Staudt,    introducing    projective    co-ordinates,    so    that 
straight  lines  and  planes  are  given  by  linear  equations.     Cay- 

85 


86  LECTURE  XI. 

ley's  theory  of  projective  measurement  leads  then  directly  to 
the  three  possible  cases  of  non-Euclidean  geometry :  hyper- 
bolic, parabolic,  and  elliptic,  according  as  the  measure  of 
curvature  k  is  <o,  =o,  or  >o.  It  is  here,  of  course,  essen- 
tial to  adopt  the  system  of  von  Staudt  and  not  that  of 
Steiner,  since  the  latter  defines  the  anharrnonic  ratio  by 
means  of  distances  of  points,  and  not  by  pure  projective 
constructions. 

(3)  Finally,  we  have  the  point  of  view  of  Riemann  and  Helm- 
holtz.  Riemann  starts  with  the  idea  of  the  element  of  distance 
ds,  which  he  assumes  to  be  expressible  in  the  form 


ds  = 

Helmholtz,  in  trying  to  find  a  reason  for  this  assumption,  con- 
siders the  motions  of  a  rigid  body  in  space,  and  derives  from 
these  the  necessity  of  giving  to  ds  the  form  indicated.  On  the 
other  hand,  Riemann  introduces  the  fundamental  notion  of  the 
measure  of  curvature  of  space. 

The  idea  of  a  measure  of  curvature  for  the  case  of  two 
variables,  i.e.  for  a  surface  in  a  three-dimensional  space,  is  due 
to  Gauss,  who  showed  that  this  is  an  intrinsic  characteristic  of 
the  surface  quite  independent  of  the  higher  space  in  which  the 
surface  happens  to  be  situated.  This  point  has  given  rise  to  a 
misunderstanding  on  the  part  of  many  non-Euclidean  writers. 
When  Riemann  attributes  to  his  space  of  three  dimensions  a 
measure  of  curvature  k,  he  only  wants  to  say  that  there  exists 
an  invariant  of  the  "form"  2d^&x&*;  he  does  not  mean  to 
imply  that  the  three-dimensional  space  necessarily  exists  as  a 
curved  space  in  a  space  of  four  dimensions.  Similarly,  the 
illustration  of  a  space  of  constant  positive  measure  of  curvature 
by  the  familiar  example  of  the  sphere  is  somewhat  misleading. 
Owing  to  the  fact  that  on  the  sphere  the  geodesic  lines  (great 
circles)  issuing  from  any  point  all  meet  again  in  another  definite 


RESEARCHES    IN    NON-EUCLIDEAN   GEOMETRY.  8/ 

point,  antipodal,  so  to  speak,  to  the  original  point,  the  existence 
of  such  an  antipodal  point  has  sometimes  been  regarded  as  a 
necessary  consequence  of  the  assumption  of  a  constant  positive 
curvature.  The  projective  theory  of  non-Euclidean  space  shows 
immediately  that  the  existence  of  an  antipodal  point,  though 
compatible  with  the  nature  of  an  elliptic  space,  is  not  necessary, 
but  that  two  geodesic  lines  in  such  a  space  may  intersect  in 
one  point  if  at  all.* 

I  call  attention  to  these  details  in  order  to  show  that  there 
is  some  advantage  in  adopting  the  second  of  the  three  points  of 
view  characterized  above,  although  the  third  is  at  least  equally 
important.  Indeed,  our  ideas  of  space  come  to  us  through  the 
senses  of  vision  and  motion,  the  "optical  properties"  of  space 
forming  one  source,  while  the  "mechanical  properties"  form 
another ;  the  former  corresponds  in  a  general  way  to  the  pro- 
jective properties,  the  latter  to  those  discussed  by  Helmholtz. 

As  mentioned  before,  from  the  point  of  view  of  projective 
geometry,  von  Staudt's  system  should  be  adopted  as  the  basis. 
It  might  be  argued  that  von  Staudt  practically  assumes  the 
axiom  of  parallels  (in  postulating  a  one-to-one  correspondence 
between  a  pencil  of  lines  and  a  row  of  points).  But  I  have 
shown  in  the  Math.  Annalen^  how  this  apparent  difficulty  can 
be  overcome  by  restricting  all  constructions  of  von  Staudt  to  a 
limited  portion  of  space. 

I  now  proceed  to  give  an  account  of  the  most  recent  re- 
searches in  non-Euclidean  geometry  made  by  Lie  and  myself. 
Lie  published  a  brief  paper  on  the  subject  in  the  Berichte  of 
the  Saxon  Academy  (1886),  and  a  more  extensive  exposition 
of  his  views  in  the  same  BericJite  for  1890  and  1891.  These 

*  This  theory  has  also  been  developed  by  Newcomb,  in  the  Journal  fur  reine 
und  angewandte  Mathematik,  Vol.  83  (1877),  pp.  293-299. 

t  Ueber  die  sogenannte  Nicht-Euklidische  Geometric,  Math.  Annalen,  Vol.  6 
(1873),  pp.  112-145. 


88  LECTURE  XI. 

papers  contain  an  application  of  Lie's  theory  of  continuous 
groups  to  the  problem  formulated  by  Helmholtz.  I  have  the 
more  pleasure  in  placing  before  you  the  results  of  Lie's  investi- 
gations as  they  are  not  taken  into  due  account  in  my  paper 
on  the  foundations  of  projective  geometry  in  Vol.  37  of  the 
Math.  Annalen  (1890),*  nor  in  my  (lithographed)  lectures  on 
non-Euclidean  geometry  delivered  at  Gottingen  in  1889-90;  the 
last  two  papers  of  Lie  appeared  too  late  to  be  considered,  while 
the  first  had  somehow  escaped  my  memory.' 

I  must  begin  by  stating  the  problem  of  Helmholtz  in  modern 
terminology.  The  motions  of  three-dimensional  space  are  oo6, 
and  form  a  group,  say  GG.  This  group  is  known  to  have  an 
invariant  for  any  two  points  /,  /',  viz.  the  distance  H  (p,  p'} 
of  these  points.  But  the  form  of  this  invariant  (and  generally 
the  form  of  the  group)  in  terms  of  the  co-ordinates  xv  x^  xz, 
y\>  y^i  y%  °f  the  P°mts  is  not  known  a  priori.  The  question 
arises  whether  the  group  of  motions  is  fully  characterized  by 
these  two  properties  so  that  none  but  the  Euclidean  and  the 
two  non-Euclidean  systems  of  geometry  are  possible. 

For  illustration  Helmholtz  made  use  of  the  analogous  case 
in  two  dimensions.  Here  we  have  a  group  of  oo3  motions  ; 
the  distance  is  again  an  invariant ;  and  yet  it  is  possible  to 
construct  a  group  not  belonging  to  any  one  of  our  three 
systems,  as  follows. 

Let  2-  be  a  complex  variable  ;  the  substitution  characterizing 
the  group  of  Euclidean  geometry  can  be  written  in  the  well- 
known  form 

z'  =  <?*>z  -\-m-\-in=  (cos  <£  +  i  sin  <f>)z  -\-m-\-  in. 

Now  modifying  this  expression  by  introducing  a  complex 
number  in  the  exponent, 

z'  =  <?<«+''%  -(-  m  -f-  in  =  ^(cos  <f>  +  i  sin  <f>)z  +  m  -f-  in, 

*  Zur  Nicht- Euklidischen  Geometrie,  pp.  544-572. 


RESEARCHES    IN    NON-EUCLIDEAN    GEOMETRY. 


89 


we  obtain  a  group  of  transformations  by  which  a  point  (in 
the  simple  case  m=o,  n  =  o)  would  not  move  about  the  origin 
in  a  circle,  but  in  a  logarithmic  spiral  ;  and  yet  this  is  a  group 
GB  with  three  variable  parameters  m,  n,  </>,  having  an  invariant 
for  every  two  points,  just  like  the  original  group.  Helmholtz 
concludes,  therefore,  that  a  new  condition,  that  of  monodromy, 
must  be  added  to  determine  our  group  completely. 

I  now  proceed  to  the  work  of  Lie.  First  as  to  the  results  : 
Lie  has  confirmed  those  of  Helmholtz  with  the  single  exception 
that  in  space  of  three  dimensions  the  axiom  of  monodromy  is 
not  needed,  but  that  the  groups  to  be  considered  are  fully 
determined  by  the  other  axioms.  As  regards  the  proofs,  how- 
ever, Lie  has  shown  that  the  considerations  of  Helmholtz  must 
be  supplemented.  The  matter  is  this.  In  keeping  one  point  of 
space  fixed,  our  GQ  will  be  reduced  to  a  G3.  Now  Helmholtz 
inquires  how  the  differentials  of  the  lines  issuing  from  the  fixed 
point  are  transformed  by  this  Gs.  For  this  purpose  he  writes 
down  the  formulae 

dxi  =  a^dx-i  +  al2dx.>  +  al3dxs, 


and  considers  the  coefficients  an,  #12,  •••  a33  as  depending  on 
three  variable  parameters.  But  Lie  remarks  that  this  is  not 
sufficiently  general.  The  linear  equations  given  above  repre- 
sent only  the  first  terms  of  power  series,  and  the  possibility 
must  be  considered  that  the  three  parameters  of  the  group  may 
not  all  be  involved  in  the  linear  terms.  In  order  to  treat  all 
possible  cases,  the  general  developments  of  Lie's  theory  of 
groups  must  be  applied,  and  this  is  just  what  Lie  does. 

Let  me  now  say  a  few  words  on  my  own  recent  researches  in 
non-Euclidean  geometry  which  will  be  found  in  a  paper  pub- 
lished in  the  Math.  Annalen,  Vol.  37  (1890),  p.  544.  Their 


cp  LECTURE   XI. 

result  is  that  our  ideas  as  to  non-Euclidean  space  are  still  very 
incomplete.  Indeed,  all  the  researches  of  Riemann,  Helmholtz, 
Lie,  consider  only  a  portion  of  space  surrounding  the  origin ; 
they  establish  the  existence  of  analytic  laws  in  the  vicinity  of 
that  point.  Now  this  space  can  of  course  be  continued,  and 
the  question  is  to  see  what  kind  of  connection  of  space  may 
result  from  this  continuation.  It  is  found  that  there  are  dif- 
ferent possibilities,  each  of  the  three  geometries  giving  rise 
to  a  series  of  subdivisions. 

To  understand  better  what  is  meant  by  these  varieties  of 
connection,  let  us  compare  the  geometry  on  a  sphere  with  that 
in  the  sheaf  of  lines  formed  by  the  diameters  of  the  sphere. 
Considering  each  diameter  as  an  infinite  line  or  ray  passing 
through  the  centre  (not  a  half-ray  issuing  from  the  centre),  to 
each  line  of  the  sheaf  there  will  correspond  two  points  on  the 
sphere,  viz.  the  two  points  of  intersection  of  the  line  with  the 
sphere.  We  have,  therefore,  a  one-to-two  correspondence 
between  the  lines  of  the  sheaf  and  the  points  of  the  sphere. 
Let  us  now  take  a  small  area  on  the  sphere ;  it  is  clear  that 
the  distance  of  two  points  contained  in  this  area  is  equal  to 
the  angle  of  the  corresponding  lines  of  the  sheaf.  Thus  the 
geometry  of  points  on  the  sphere  and  the  geometry  of  lines  in 
the  sheaf  are  identical  as  far  as  small  regions  are  concerned,  both 
corresponding  to  the  assumption  of  a  constant  positive  measure 
of  curvature.  A  difference  appears,  however,  as  soon  as  we 
consider  the  whole  closed  sphere  on  the  one  hand  and  the  com- 
plete sheaf  on  the  other.  Let  us  take,  for  instance,  two  geodesic 
lines  of  the  sphere,  i.e.  two  great  circles,  which  evidently  inter- 
sect in  two  (diametral)  points.  The  corresponding  pencils  of 
the  sheaf  have  only  one  straight  line  in  common. 

A  second  example  for  this  distinction  occurs  in  comparing 
the  geometry  of  the  Euclidean  plane  with  the  geometry  on  a 
closed  cylindrical  surface.  The  latter  can  be  developed  in  the 


RESEARCHES    IN    NON-EUCLIDEAN    GEOMETRY.  91 

usual  way  into  a  strip  of  the  plane  bounded  by  two  parallel 
lines,  as  will  appear  from  Fig.  20,  the  arrows  indicating  that 
the  opposite  points  of  the  edges  are  coincident  on  the  cylin- 
drical surface.  We  notice  at  once  the  difference  :  while  in  the 
plane  all  geodesic  lines  are  infinite,  on  the  cylinder  there  is 


Fig.  20. 

one  geodesic  line  that  is  of  finite  length,  and  while  in  the  plane 
two  geodesic  lines  always  intersect  in  one  point,  if  at  all,  on 
the  cylinder  there  may  be  oo  points  of  intersection. 

This    second    example   was   generalized    by   Clifford   in   an 
address  before  the  Bradford  meeting  of  the  British  Associa- 


Fig.  21. 


tion  (1873).  In  accordance  with  Clifford's  general  idea,  we 
may  define  a  closed  surface  by  taking  a  parallelogram  out  of 
an  ordinary  plane  and  making  the  opposite  edges  correspond 
point  to  point  as  indicated  in  Fig.  21.  It  is  not  to  be 
understood  that  the  opposite  edges  should  be  brought  to 


92  LECTURE  XI. 

coincidence  by  bending  the  parallelogram  (which  evidently 
would  be  impossible  without  stretching) ;  but  only  the  logical 
convention  is  made  that  the  opposite  points  should  be  con- 
sidered as  identical.  Here,  then,  we  have  a  closed  mani- 
foldness  of  the  connectivity  of  an  anchor-ring,  and  every  one 
will  see  the  great  differences  that  exist  here  in  comparison 
with  the  Euclidean  plane  in  everything  concerning  the  lengths 
and  the  intersections  of  geodesic  lines,  etc. 

It  is  interesting  to  consider  the  G3  of  Euclidean  motions  on 
this  surface.  There  is  no  longer  any  possibility  of  moving  the 
surface  on  itself  in  oo3  ways,  the  closed  surface  being  consid- 
ered in  its  totality.  But  there  is  no  difficulty  in  moving  any 
small  area  over  the  closed  surface  in  oo3  ways. 

We  have  thus  found,  in  addition  to  the  Euclidean  plane, 
two  other  forms  of  surfaces  :  the  strip  between  parallels  and 
Clifford's  parallelogram.  Similarly  we  have  by  the  side  of 
ordinary  Euclidean  space  three  other  types  with  the  Euclid- 
ean element  of  arc ;  one  of  these  results  from  considering  a 
parallelepiped. 

Here  I  introduce  the  axiomatic  element.  There  is  no  way 
of  proving  that  the  whole  of  space  can  be  moved  in  itself  in 
oo6  ways ;  all  we  know  is  that  small  portions  of  space  can  be 
moved  in  space  in  oo6  ways.  Hence  there  exists  the  possibility 
that  our  actual  space,  the  measure  of  curvature  being  taken  as 
zero,  may  correspond  to  any  one  of  the  four  cases. 

Carrying  out  the  same  considerations  for  the  spaces  of  con- 
stant positive  measure  of  curvature,  we  are  led  back  to  the  two 
cases  of  elliptic  and  spherical  geometry  mentioned  before.  If, 
however,  the  measure  of  curvature  be  assumed  as  a  negative 
constant,  we  obtain  an  infinite  number  of  cases,  corresponding 
exactly  to  the  configurations  considered  by  Poincar6  and  myself 
in  the  theory  of  automorphic  functions.  This  I  shall  not  stop 
to  develop  here. 


RESEARCHES    IN   NON-EUCLIDEAN    GEOMETRY. 


93 


I  may  add  that  Killing  has  verified  this  whole  theory.*  It 
is  evident  that  from  this  point  of  view  many  assertions  con- 
cerning space  made  by  previous  writers  are  no  longer  correct 
(e.g.  that  infinity  of  space  is  a  consequence  of  zero  curvature), 
so  that  we  are  forced  to  the  opinion  that  our  geometrical 
demonstrations  have  no  absolute  objective  truth,  but  are  true 
only  for  the  present  state  of  our  knowledge.  These  demon- 
strations are  always  confined  within  the  range  of  the  space- 
conceptions  that  are  familiar  to  us  ;  and  we  can  never  tell 
whether  an  enlarged  conception  may  not  lead  to  further 
possibilities  that  would  have  to  be  taken  into  account. 
From  this  point  of  view  we  are  led  in  geometry  to  a  certain 
modesty,  such  as  is  always  in  place  in  the  physical  sciences. 

*  Ueber  die  Clifford-Klein 'schen  Raumformcn,  Math.  Annalen,  Vol.  39  (1891), 
pp.  257-278. 


LECTURE  XII.:    THE   STUDY    OF   MATHEMATICS 
AT   GOTTINGEN. 

(September  9,  1893.) 

IN  this  last  lecture  I  should  like  to  make  some  general 
remarks  on  the  way  in  which  the  study  of  mathematics  is 
organized  at  the  university  of  Gottingen,  with  particular  refer- 
ence to  what  may  be  of  interest  to  American  students.  At  the 
same  time  I  desire  to  give  you  an  opportunity  to  ask  any  ques- 
tions that  may  occur  to  you  as  to  the  broader  subject  of  mathe- 
matical study  at  German  universities  in  general.  I  shall  be 
glad  to  answer  such  inquiries  to  the  extent  of  my  ability. 

It  is  perhaps  inexact  to  speak  of  an  organization  of  the 
mathematical  teaching  at  Gottingen  ;  you  know  that  Lern-  und 
Lehr-Freiheit  prevail  at  a  German  university,  so  that  the  organ- 
ization I  have  in  mind  consists  merely  in  a  voluntary  agreement 
among  the  mathematical  professors  and  instructors.  We  dis- 
tinguish at  Gottingen  between  a  general  and  a  higher  course 
in  mathematics.  The  general  course  is  intended  for  that  large 
majority  of  our  students  whose  intention  it  is  to  devote  them- 
selves to  the  teaching  of  mathematics  and  physics  in  the  higher 
schools  (Gymnasien,  Realgymnasien,  Realschuleri),  while  the 
higher  course  is  designed  specially  for  those  whose  final  aim 
is  original  investigation. 

As  regards  the  former  class  of  students,  it  is  my  opinion  that 
in  Germany  (here  in  America,  I  presume,  the  conditions  are 
very  different)  the  abstractly  theoretical  instruction  given  to 

94 


THE  STUDY  OF  MATHEMATICS  AT  GOTTINGEN. 


95 


them  has  been  carried  too  far.  It  is  no  doubt  true  that  what 
the  university  should  give  the  student  above  all  other  things 
is  the  scientific  ideal.  For  this  reason  even  these  students 
should  push  their  mathematical  studies  far  beyond  the  elemen- 
tary branches  they  may  have  to  teach  in  the  future.  But  the 
ideal  set  before  them  should  not  be  chosen  so  far  distant,  and 
so  out  of  connection  with  their  more  immediate  wants,  as  to 
make  it  difficult  or  impossible  for  them  to  perceive  the  bear- 
ing that  this  ideal  has  on  their  future  work  in  practical  life. 
In  other  words,  the  ideal  should  be  such  as  to  fill  the  future 
teacher  with  enthusiasm  for  his  life-work,  not  such  as  to  make 
him  look  upon  this  work  with  contempt  as  an  unworthy 
drudgery. 

For  this  reason  we  insist  that  our  students  of  this  class,  in 
addition  to  their  lectures  on  pure  mathematics,  should  pursue 
a  thorough  course  in  physics,  this  subject  forming  an  integral 
part  of  the  curriculum  of  the  higher  schools.  Astronomy  is 
also  recommended  as  showing  an  important  application  of 
mathematics  ;  and  I  believe  that  the  technical  branches,  such 
as  applied  mechanics,  resistance  of  materials,  etc.,  would  form 
a  valuable  aid  in  showing  the  practical  bearing  of  mathematical 
science.  Geometrical  drawing  and  descriptive  geometry  form 
also  a  portion  of  the  course.  Special  exercises  in  the  solution 
of  problems,  in  lecturing,  etc.,  are  arranged  in  connection  with 
the  mathematical  lectures,  so  as  to  bring  the  students  into 
personal  contact  with  the  instructors. 

I  wish,  however,  to  speak  here  more  particularly  on  the 
higher  courses,  as  these  are  of  more  special  interest  to  Ameri- 
can students.  Here  specialization  is  of  course  necessary. 
Each  professor  and  decent  delivers  certain  lectures  specially 
designed  for  advanced  students,  in  particular  for  those  studying 
for  the  doctor's  degree.  Owing  to  the  wide  extent  of  modern 
mathematics,  it  would  be  out  of  the  question  to  cover  the  whole 


96  LECTURE  XII. 

field.  These  lectures  are  therefore  not  regularly  repeated  every 
year ;  they  depend  largely  on  the  special  line  of  research  that 
happens  at  the  time  to  engage  the  attention  of  the  professor. 
In  addition  to  the  lectures  we  have  the  higher  seminaries,  whose 
principal  object  is  to  guide  the  student  in  original  investigation 
and  give  him  an  opportunity  for  individual  work. 

As  regards  my  own  higher  lectures,  I  have  pursued  a  certain 
plan  in  selecting  the  subjects  for  different  years,  my  general 
aim  being  to  gain,  in  the  course  of  time,  a  complete  view  of  the 
whole  field  of  modern  mathematics,  with  particular  regard  to  the 
intuitional  or  (in  the  highest  sense  of  the  term)  geometrical 
standpoint.  This  general  tendency  you  will,  I  trust,  also  find 
expressed  in  this  colloquium,  in  which  I  have  tried  to  present, 
within  certain  limits,  a  general  programme  of  my  individual 
work.  To  carry  out  this  plan  in  Gottingen,  and  to  bring  it  to 
the  notice  of  my  students,  I  have,  for  many  years,  adopted  the 
method  of  having  my  higher  lectures  carefully  written  out,  and, 
in  recent  years,  of  having  them  lithographed,  so  as  to  make 
them  more  readily  accessible.  These  former  lectures  are  at  the 
disposal  of  my  hearers  for  consultation  at  the  mathematical 
reading-room  of  the  university ;  those  that  are  lithographed  can 
be  acquired  by  anybody,  and  I  am  much  pleased  to  find  them 
so  well  known  here  in  America. 

As  another  important  point,  I  wish  to  say  that  I  have  always 
regarded  my  students  not  merely  as  hearers  or  pupils,  but  as 
collaborators.  I  want  them  to  take  an  active  part  in  my  own 
researches  ;  and  they  are  therefore  particularly  welcome  if  they 
bring  with  them  special  knowledge  and  new  ideas,  whether 
these  be  original  with  them,  or  derived  from  some  other  source, 
from  the  teachings  of  other  mathematicians.  Such  men  will 
spend  their  time  at  Gottingen  most  profitably  to  themselves. 

I  have  had  the  pleasure  of  seeing  many  Americans  among 
my  students,  and  gladly  bear  testimony  to  their  great  enthusi- 


THE  STUDY  OF  MATHEMATICS  AT  GOTTINGEN. 


97 


asm  and  energy.  Indeed,  I  do  not  hesitate  to  say  that,  for 
some  years,  my  higher  lectures  were  mainly  sustained  by  stu- 
dents whose  home  is  in  this  country.  But  I  deem  it  my  duty 
to  refer  here  to  some  difficulties  that  have  occasionally  arisen 
in  connection  with  the  coming  of  American  students  to  Gottin- 
gen.  Perhaps  a  frank  statement  on  my  part,  at  this  opportunity, 
will  contribute  to  remove  these  difficulties  in  part.  What  I  wish 
to  speak  of  is  this.  It  frequently  happens  at  Gottingen,  and 
probably  at  other  German  universities  as  well,  that  American 
students  desire  to  take  the  higher  courses  when  their  prepara- 
tion is  entirely  inadequate  for  such  work.  A  student  having 
nothing  but  an  elementary  knowledge  of  the  differential  and 
integral  calculus,  usually  coupled  with  hardly  a  moderate  famil- 
iarity with  the  German  language,  makes  a  decided  mistake  in 
attempting  to  attend  my  advanced  lectures.  If  he  comes  to  Got- 
tingen with  such  a  preparation  (or,  rather,  the  lack  of  it),  he 
may,  of  course,  enter  the  more  elementary  courses  offered  at  our 
university ;  but  this  is  generally  not  the  object  of  his  coming. 
Would  he  not  do  better  to  spend  first  a  year  or  two  in  one  of 
the  larger  American  universities  ?  Here  he  would  find  more 
readily  the  transition  to  specialized  studies,  and  might,  at  the 
same  time,  arrive  at  a  clearer  judgment  of  his  own  mathematical 
ability ;  this  would  save  him  from  the  severe  disappointment 
that  might  result  from  his  going  to  Germany. 

I  trust  that  these  remarks  will  not  be  misunderstood.  My 
presence  here  among  you  is  proof  enough  of  the  value  I  attach 
to  the  coming  of  American  students  to  Gottingen.  It  is  in 
the  interest  of  those  wishing  to  go  there  that  I  speak ;  and 
for  this  reason  I  should  be  glad  to  have  the  widest  publicity 
given  to  what  I  have  said  on  this  point. 

Another  difficulty  lies  in  the  fact  that  my  higher  lectures 
have  frequently  an  encyclopedic  character,  conformably  to  the 
general  tendency  of  my  programme.  This  is  not  always  just 


98  LECTURE   XII. 

what  is  most  needful  to  the  American  student,  whose  work 
is  naturally  directed  to  gaining  the  doctor's  degree.  He  will 
need,  in  addition  to  what  he  may  derive  from  my  lectures,  the 
concentration  on  a  particular  subject ;  and  this  he  will  often 
find  best  with  other  instructors,  at  Gottingen  or  elsewhere. 
I  wish  to  state  distinctly  that  I  do  not  regard  it  as  at  all  desira- 
ble that  all  students  should  confine  their  mathematical  studies 
to  my  courses  or  even  to  Gottingen.  On  the  contrary,  it 
seems  to  me  far  preferable  that  the  majority  of  the  students 
should  attach  themselves  to  other  mathematicians  for  certain 
special  lines  of  work.  My  lectures  may  then  serve  to  form 
the  wider  background  on  which  these  special  studies  are  pro- 
jected. It  is  in  this  way,  I  believe,  that  my  lectures  will 
prove  of  the  greatest  benefit. 

In  concluding  I  wish  to  thank  you  for  your  kind  attention, 
and  to  give  expression  to  the  pleasure  I  have  found  in  meeting 
here  at  Evanston,  so  near  to  Chicago,  the  great  metropolis  of 
this  commonwealth,  a  number  of  enthusiastic  devotees  of  my 
chosen  science. 


THE   DEVELOPMENT  OF  MATHEMATICS  AT   THE 
GERMAN   UNIVERSITIES.* 

BY  F.  KLEIN. 

THE  eighteenth  century  laid  the  firm  foundation  for  the 
development  of  mathematics  in  all  directions.  The  universi- 
ties as  such,  however,  did  not  take  a  prominent  part  in  this 
work ;  the  academies  must  here  be  considered  of  prime  impor- 
tance. Nor  can  any  fixed  limits  of  nationality  be  recognized. 
At  the  beginning  of  the  period  there  appears  in  Germany  no 
less  a  man  than  Leibniz;  then  follow,  among  the  kindred 
Swiss,  the  dynasty  of  the  Bernoullis  and  the  incomparable 
Elder.  But  the  activity  of  these  men,  even  in  its  outward 
manifestation,  was  not  confined  within  narrow  geographical 
bounds ;  to  encompass  it  we  must  include  the  Netherlands, 
and  in  particular  Russia,  with  Germany  and  Switzerland.  On 
the  other  hand,  under  Frederick  the  Great,  the  most  eminent 
French  mathematicians,  Lagrange,  d'Alembert,  Maupertuis, 
formed  side  by  side  with  Euler  and  Lambert  the  glory  of 
the  Berlin  Academy.  The  impulse  toward  a  complete  change 
in  these  conditions  came  from  the  French  Revolution. 

The  influence  of  this  great  historical  event  on  the  devel- 
opment of  science  has  manifested  itself  in  two  directions. 
On  the  one  hand  it  has  effected  a  wider  separation  of  nations 

*  Translation,  with  a  few  slight  modifications  by  the  author,  of  the  section  Mathe- 
matik  in  the  work  Die  deutschen  Universitaten,  Berlin,  A.  Asher  &  Co.,  1893, 
prepared  by  Professor  Lexis  for  the  World's  Columbian  Exposition  at  Chicago. 

99 


100  THE   DEVELOPMENT   OF   MATHEMATICS 

with  a  distinct  development  of  characteristic  national  quali- 
ties. Scientific  ideas  preserve,  of  course,  their  universality ; 
indeed,  international  intercourse  between  scientific  men  has 
become  particularly  important  for  the  progress  of  science ; 
but  the  cultivation  and  development  of  scientific  thought  now 
progress  on  national  bases.  The  other  effect  of  the  French 
Revolution  is  in  the  direction  of  educational  methods.  The 
decisive  event  is  the  foundation  of  the  Ecole  polytechnique  at 
Paris  in  1794.  That  scientific  research  and  active  instruction 
can  be  directly  combined,  that  lectures  alone  are  not  suffi- 
cient, and  must  be  supplemented  by  direct  personal  intercourse 
between  the  lecturer  and  his  students,  that  above  all  it  is  of 
prime  importance  to  arouse  the  student's  own  activity,  —  these 
are  the  great  principles  that  owe  to  this  source  their  recogni- 
tion and  acceptance.  The  example  of  Paris  has  been  the  more 
effective  in  this  direction  as  it  became  customary  to  publish  in 
systematic  form  the  lectures  delivered  at  this  institution  ;  thus 
arose  a  series  of  admirable  text-books  which  remain  even  now 
the  foundation  of  mathematical  study  everywhere  in  Germany. 
Nevertheless,  the  principal  idea  kept  in  view  by  the  founders 
of  the  Polytechnic  School  has  never  taken  proper  root  in  the 
German  universities.  This  is  the  combination  of  the  technical 
with  the  higher  mathematical  training.  It  is  true  that,  prima- 
rily, this  has  been  a  distinct  advantage  for  the  unrestricted 
development  of  theoretical  investigation.  Our  professors,  find- 
ing themselves  limited  to  a  small  number  of  students  who,  as 
future  teachers  and  investigators,  would  naturally  take  great 
interest  in  matters  of  pure  theory,  were  able  to  follow  the  bent 
of  their  individual  predilections  with  much  greater  freedom 
than  would  have  been  possible  otherwise. 

But  we  anticipate  our  historical  account.  First  of  all  we 
must  characterize  the  position  that  Gauss  holds  in  the  science 
of  this  age.  Gauss  stands  in  the  very  front  of  the  new  develop- 


AT   THE   GERMAN   UNIVERSITIES.  IOI 

ment :  first,  by  the  time  of  his  activity,  his  publications  reach- 
ing back  to  the  year  1799,  and  extending  throughout  the  entire 
first  half  of  the  nineteenth  century  ;  then  again,  by  the  wealth  of 
new  ideas  and  discoveries  that  he  has  brought  forward  in  almost 
every  branch  of  pure  and  applied  mathematics,  and  which  still 
preserve  their  fruitfulness  ;  finally,  by  his  methods,  for  Gauss 
was  the  first  to  restore  that  rigour  of  demonstration  which  we 
admire  in  the  ancients,  and  which  had  been  forced  unduly  into 
the  background  by  the  exclusive  interest  of  the  preceding  period 
in  new  developments.  And  yet  I  prefer  to  rank  Gauss  with 
the  great  investigators  of  the  eighteenth  century,  with  Euler, 
Lagrange,  etc.  H^Jhelongs  to  them  by  the  universality  of  his 
work,  in  which  no  trace' £s  yet  appears  of  that  specialization 
which  has  become  the  characteristic  of  our  times.  He  belongs 


to  them  by  his  exclusively  acaden?^  interest,  by  the  absence  of 
the  modern  teaching  activity  just  char/C^erized.  We  shall  have 
a  picture  of  the  development  of  mathematics  if  we  imagine  a 
chain  of  lofty  mountains  as  representative  of  the  men  of  the 
eighteenth  century,  terminating  in  a  mighty  outlying  summit, — 
Gauss,  —  and  then  a  broader,  hilly  country  of  lower  elevation  ; 
but  teeming  with  new  elements  of  life.  More  immediately  con- 
nected with  Gauss  we  find  in  the  following  period  only  the 
astronomers  and  geodesists  under  the  dominating  influence  of 
Bessel ;  while  in  theoretical  mathematics,  as  it  begins  hence- 
forth to  be  independently  cultivated  in  our  universities,  a  new 
epoch  begins  with  the  second  quarter  of  the  present  century, 
marked  by  the  illustrious  names  oijacobi  and  Dirichlet. 

Jacobi  came  originally  from  Berlin  and  returned  there  for 
the  closing  years  of  his  life  (died  1851).  But  it  is  the  period 
from  1826  to  1843,  when  he  worked  at  Konigsberg  with  Bessel 
and  Franz  Neumann,  that  must  be  regarded  as  the  culmination 
of  his  activity.  There  he  published  in  1829  his  Fundamenta 
nova  theories  functionum  ellipticarum,  in  which  he  gave,  in 


102  THE   DEVELOPMENT   OF   MATHEMATICS 

analytic  form,  a  systematic  exposition  .of  his  own  discoveries 
and  those  of  Abel  in  this  field.  Then  followed  a  prolonged  resi- 
dence in  Paris,  and  finally  that  remarkable  activity  as  a  teacher, 
which  still  remains  without  a  parallel  in  stimulating  power  as 
well  as  in  direct  results  in  the  field  of  pure  mathematics.  An 
idea  of  this  work  can  be  derived  from  the  lectures  on  dynamics, 
edited  by  Clebsch  in  1866,  and  from  the  complete  list  of  his 
Konigsberg  lectures  as  compiled  by  Kronecker  in  the  seventh 
volume  of  the  Gesammelte  Werke.  The  new  feature  is  that 
Jacobi  lectured  exclusively  on  those  problems  on  which  he  was 
working  himself,  and  made  it  his  sole  object  to  introduce  his 
students  into  his  own  circle  of  ideas.  With  this  end  in  view 
he  founded,  for  instance,  the  first  mathematical  seminary.  And 
so  great  was  his  enthusiasm  that  often  he  not  only  gave  the 
most  important  new  results  of  his  researches  in  these  lectures, 
but  did  not  even  take  the  time  to  publish  them  elsewhere. 

Dirichlet  worked  first  in  Breslau,  then  for  a  long  period 
(1831-1855)  in  Berlin,  and  finally  for  four  years  in  Gottingen. 
Following  Gauss,  but  at  the  same  time  in  close  connection 
with  the  contemporary  French  scholars,  he  chose  mathemati- 
cal physics  and  the  theory  of  numbers  as  the  central  points 
of  his  scientific  activity.  It  is  to  be  noticed  that  his  interest  is 
directed  less  towards  comprehensive  developments  than  towards 
simplicity  of  conception  and  questions  of  principle ;  these  are 
also  the  considerations  on  which  he  insists  particularly  in  his 
lectures.  These  lectures  are  characterized  by  perfect  lucidity 
and  a  certain  refined  objectivity ;  they  are  at  the  same  time 
particularly  accessible  to  the  beginner  and  suggestive  in  a  high 
degree  to  the  more  advanced  reader.  It  may  be  sufficient  to 
refer  here  to  his  lectures  on  the  theory  of  numbers,  edited  by 
Dedekind  ;  they  still  form  the  standard  text-book  on  this  subject. 

With  Gauss,  Jacobi,  Dirichlet,  we  have  named  the  men  who 
have  determined  the  direction  of  the  subsequent  development. 


AT   THE   GERMAN   UNIVERSITIES.  IO3 

We  shall  now  continue  our  account  in  a  different  manner, 
arranging  it  according  to  the  universities  that  have  been  most 
prominent  from  a  mathematical  standpoint.  For  henceforth, 
besides  the  special  achievements  of  individual  workers,  the 
principle  of  co-operation,  with  its  dependence  on  local  condi- 
tions, comes  to  have  more  and  more  influence  on  the  advance- 
ment of  our  science.  Setting  the  upper  limit  of  our  account 
about  the  year  1870,  we  may  name  the  universities  of  Konigs- 
berg, Berlin,  Gottingen,  and  Heidelberg. 

Of  Jacobi's  activity  at  Konigsberg  enough  has  already  been 
said.  It  may  now  be  added  that  even  after  his  departure  the 
university  remained  a  centre  of  mathematical  instruction. 
Richelot  and  Hesse  knew  how  to  maintain  the  high  tradition  of 
Jacobi,  the  former  on  the  analytical,  the  latter  on  the  geomet- 
rical side.  At  the  same  time  Franz  Neumann  s  lectures  on 
mathematical  physics  began  to  attract  more  and  more  atten- 
tion. A  stately  procession  of  mathematicians  has  come  from 
Konigsberg ;  there  is  scarcely  a  university  in  Germany  to 
which  Konigsberg  has  not  sent  a  professor. 

Of  Berlin,  too,  we  have  already  anticipated  something  in  our 
account.  The  years  from  1845  to  1851,  during  which  Jacobi 
and  Dirichlet  worked  together,  form  the  culminating  period  of 
the  first  Berlin  school.  Besides  these  men  the  most  promi- 
nent figure  is  that  of  Steiner  (connected  with  the  university 
from  1835  to  1864),  the  founder  of  the  German  synthetic 
geometry.  An  altogether  original  character,  he  was  a  highly 
effective  teacher,  owing  to  the  one-sidedness  with  which  he 
developed  his  geometrical  conceptions. — As  an  event  of  no 
mean  importance,  we  must  here  record  the  foundation  (in  1826) 
of  Crelle  s  Journal  fur  reine  und  angewandte  Mathematik.  This, 
for  decades  the  only  German  mathematical  periodical,  contained 
in  its  pages  the  fundamental  memoirs  of  nearly  all  the  emi- 
nent representatives  of  the  rapidly  growing  science  in  Germany. 


104  THE   DEVELOPMENT   OF   MATHEMATICS 

Among  foreign  contributions  the  very  first  volumes  presented 
Abel's  pioneer  researches.  Crelle  himself  conducted  this  peri- 
odical for  thirty  years;  then  followed  Borchardt,  1856-1880; 
now  the  Journal  has  reached  its  uoth  volume.  —  We  must 
also  mention  the  formation  (in  1844)  of  the  Berliner  physi- 
kalische  Gesellschaft.  Men  like  Helmholtz,  Kirclihoff,  and 
Clansius  have  grown  up  here  ;  and  while  these  men  cannot 
be  assigned  to  mathematics  in  the  narrower  sense,  their  work 
has  been  productive  of  important  results  for  our  science  in 
various  ways.  During  the  same  period,  Encke  exercised,  as 
director  of  the  Berlin  astronomical  observatory  (1825-1862), 
a  far-reaching  influence  by  elaborating  the  methods  of  astro- 
nomical calculation  on  the  lines  first  laid  down  by  Gauss.  — 
We  leave  Berlin  at  this  point,  reserving  for  the  present  the 
account  of  the  more  recent  development  of  mathematics  at 
this  university. 

The  discussion  of  the  Gottingen  school  will  here  find  its 
appropriate  place.  The  permanent  foundation  on  which  the 
mathematical  importance  of  Gottingen  rests  is  necessarily  the 
Gauss  tradition.  This  found,  indeed,  its  direct  continuation 
on  the  physical  side  when  Wilhelm  Weber  returned  from 
Leipsic  to  Gottingen  (1849)  and  for  the  first  time  established 
systematic  exercises  in  those  methods  of  exact  electro-magnetic 
measurement  that  owed  their  origin  to  Gauss  and  himself. 
On  the  mathematical  side  several  eminent  names  follow  in 
rapid  succession.  After  Gauss's  death,  Dirichlet  was  called 
as  his  successor  and  transferred  his  great  activity  as  a  teacher 
to  Gottingen,  for  only  too  brief  a  period  (1855-59).  By  his 
side  grew  up  Riemann  (1854-66),  to  be  followed  later  by 
Clebsch  (1868-72). 

Riemann  takes  root  in  Gauss  and  Dirichlet ;  on  the  other 
hand  he  fully  assimilated  Cauchy's  ideas  as  to  the  use  of 
complex  variables.  Thus  arose  his  profound  creations  in  the 


AT   THE   GERMAN   UNIVERSITIES. 


105 


theory  of  functions  which  ever  since  have  proved  a  rich  and 
permanent  source  of  the  most  suggestive  material.  Clebsch 
sustains,  so  to  speak,  a  complementary  relation  to  Riemann. 
Coming  originally  from  Konigsberg,  and  occupied  with  mathe- 
matical physics,  he  had  found  during  the  period  of  his  work 
at  Giessen  (1863-68)  the  particular  direction  which  he  after- 
wards followed  so  successfully  at  Gottingen.  Well  acquainted 
with  the  work  of  Jacobi  and  with  modern  geometry,  he  intro- 
duced into  these  fields  the  results  of  the  algebraic  researches  of 
the  English  mathematicians  Cayley  and  Sylvester,  and  on  the 
double  foundation  thus  constructed,  proceeded  to  build  up  new 
approaches  to  the  problems  of  the  entire  theory  of  functions, 
and  in  particular  to  Riemann's  own  developments.  But  with 
this  the  significance  of  Clebsch  for  the  development  of  our 
science  is  not  completely  characterized.  A  man  of  vivid  imagi- 
nation who  readily  entered  into  the  ideas  of  others,  he  influ- 
enced his  students  far  beyond  the  limits  of  direct  instruction  ; 
of  an  active  and  enterprising  character,  he  founded,  together 
with  C.  Neumann  in  Leipsic,  a  new  periodical,  the  Mathe- 
matische  Annalen,  which  has  since  been  regularly  continued, 
and  is  just  concluding  its  41  st  volume. 

We  recall  further  those  memorable  years  of  Heidelberg,  from 
1855  to  perhaps  1870.  Here  were  delivered  Hesse's  elegant 
and  widely  read  lectures  on  analytic  geometry.  Here  Kirch- 
hoff  produced  his  lectures  on  mathematical  physics.  Here, 
above  all,  Helmholtz  completed  his  great  papers  on  mathe- 
matical physics,  which  in  their  turn  served  as  basis  for  Kirch- 
hoff's  elegant  later  researches. 

It  remains  now  to  speak  of  the  second  Berlin  school,  beginning 
also  about  the  middle  of  the  century,  but  still  operating  upon 
the  present  age.  Kummer,  Kronecker,  Weierstrass,  have  been 
its  leaders,  the  first  two,  as  students  of  Dirichlet,  pre-eminently 
engaged  in  developing  the  theory  of  numbers,  while  the  last, 


106  THE   DEVELOPMENT   OF   MATHEMATICS 

leaning  more  on  Jacobi  and  Cauchy,  became,  together  with 
Riemann,  the  creator  of  the  modern  theory  of  functions. 
Kummer's  lectures  can  here  merely  be  named  in  passing ; 
with  their  clear  arrangement  and  exposition  they  have  always 
proved  especially  useful  to  the  majority  of  students,  without 
being  particularly  notable  for  their  specific  contents.  Quite 
different  is  the  case  of  Kronecker  and  Weierstrass,  whose 
lectures  became  in  the  course  of  time  more  and  more  the 
expression  of  their  scientific  individuality.  To  a  certain  ex- 
tent both  have  thrust  intuitional  methods  into  the  back- 
ground and,  on  the  other  hand,  have  in  a  measure  avoided 
the  long  formal  developments  of  our  science,  applying  them- 
selves with  so  much  the  keener  criticism  to  the  fundamental 
analytical  ideas.  In  this  direction  Kronecker  has  gone  even 
farther  than  Weierstrass  in  trying  to  banish  altogether  the 
idea  of  the  irrational  number,  and  to  reduce  all  developments 
to  relations  between  integers  alone.  The  tendencies  thus 
characterized  have  exerted  a  wide-felt  influence,  and  give  a 
distinctive  character  to  a  large  part  of  our  present  mathe- 
matical investigations. 

We  have  thus  sketched  in  general  outlines  the  state  reached 
by  our  science  about  the  year  1870.  It  is  impossible  to  carry 
our  account  beyond  this  date  in  a  similar  form.  For  the  devel- 
opments that  now  arise  are  not  yet  finished ;  the  persons  whom 
we  should  have  to  name  are  still  in  the  midst  of  their  creative 
activity.  All  we  can  do  is  to  add  a  few  remarks  of  a  more 
general  nature  on  the  present  aspect  of  mathematical  science 
in  Germany.  Before  doing  this,  however,  we  must  supple- 
ment the  preceding  account  in  two  directions. 

Let  it  above  all  be  emphasized  that  even  within  the  limits 
here  chosen,  we  have  by  no  means  exhausted  the  subject.  It 
is,  indeed,  characteristic  of  the  German  universities  that  their 
life  is  not  wholly  centralized,  — that  wherever  a  leader  appears, 


AT   THE   GERMAN   UNIVERSITIES. 


107 


he  will  find  a  sphere  of  activity.  We  may  name  here,  from  an 
earlier  period,  the  acute  analyst  J.  Fr.  Pfaff,  who  worked  in 
Helmstadt  and  Halle  from  1788  to  1825,  and,  at  one  time,  had 
Gauss  among  his  students.  Pfaff  was  the  first  representative 
of  the  combinatory  school,  which,  for  a  time,  played  a  great  r61e 
in  different  German  universities,  but  was  finally  pushed  aside  in 
the  manifold  development  of  the  advancing  science.  We  must 
further  mention  the  three  great  geometers,  Mbbius  in  Leipsic, 
Pliicker  in  Bonn,  von  Staudt  in,  Erlangen.  Mobius  was,  at  the 
same  time,  an  astronomer,  and  conducted  the  Leipsic  observa- 
tory from  1816  till  1868.  Pliicker,  again,  devoted  only  the  first 
half  of  his  productive  period  (1826-46)  to  mathematics,  turning 
his  attention  later  to  experimental  physics  (where  his  researches 
are  well  known),  and  only  returning  to  geometrical  investigation 
towards  the  close  of  his  life  (1864-68).  The  accidental  circum- 
stance that  each  of  these  three  men  worked  as  teacher  only  in 
a  narrow  circle  has  kept  the  development  of  modern  geometry 
unduly  in  the  background  in  our  sketch.  Passing  beyond 
university  circles,  we  may  be  allowed  to  add  the  name  of 
Grassmann,  of  Stettin,  who,  in  his  Ausdehnungslehre  (1844  an^ 
1862),  conceived  a  system  embracing  the  results  of  modern 
geometrical  speculation,  and,  from  a  very  different  field,  that  of 
Hansen,  of  Gotha,  the  celebrated  representative  of  theoretical 
astronomy. 

We  must  also  mention,  in  a  few  words,  the  development  of 
technical  education.  About  the  middle  of  the  century,  it  became 
the  custom  to  call  mathematicians  of  scientific  eminence  to  the 
polytechnic  schools.  Foremost  in  this  respect  stands  Zurich, 
which,  in  spite  of  the  political  boundaries,  may  here  be  counted 
as  our  own  ;  indeed,  quite  a  number  of  professors  have  taught 
in  the  Zurich  polytechnic  school  who  are  to-day  ornaments  of 
the  German  universities.  Thus  the  ideal  of  the  Paris  school, 
the  combination  of  mathematical  with  technical  education, 


108  THE   DEVELOPMENT   OF   MATHEMATICS 

became  again  more  prominent.  A  considerable  influence  in 
this  direction  was  exercised  by  Redtenbacher  s  lectures  on  the 
theory  of  machines  which  attracted  to  Carlsruhe  an  ever-increas- 
ing number  of  enthusiastic  students.  Descriptive  geometry  and 
kinematics  were  scientifically  elaborated.  Culmann  of  Zurich, 
in  creating  graphical  statics,  introduced  the  principles  of  modern 
geometry,  in  the  happiest  manner,  into  mechanics.  In  connec- 
tion with  the  scientific  advance  thus  outlined,  numerous  new 
polytechnic  schools  were  founded  in  Germany  about  1870  and 
during  the  following  years,  and  some  of  the  older  schools  were 
reorganized.  At  Munich  and  Dresden,  in  particular,  in  accord- 
ance with  the  example  of  Zurich,  special  departments  for  the 
training  of  teachers  and  professors  were  established.  The 
polytechnic  schools  have  thus  attained  great  importance  for 
mathematical  education  as  well  as  for  the  advancement  of  the 
science.  We  must  forbear  to  pursue  more  closely  the  many 
interesting  questions  that  present  themselves  in  this  connection. 

If  we  survey  the  entire  field  of  development  described  above, 
this,  at  any  rate,  appears  as  the  obvious  conclusion,  in  Germany 
as  elsewhere,  that  the  number  of  those  who  have  an  earnest 
interest  in  mathematics  has  increased  very  rapidly  and  that,  as  a 
consequence,  the  amount  of  mathematical  production  has  grown 
to  enormous  proportions.  In  this  respect  an  imperative  need 
was  supplied  when  Ohrtmann  and  Mutter  established  in  Berlin 
(1869)  an  annual  bibliographical  review,  Die  Fortschritte  der 
Mathematik,  of  which  the  2ist  volume  has  just  appeared. 

In  conclusion  a  few  words  should  here  be  said  concerning  the 
modern  development  of  university  instruction.  The  principal 
effort  has  been  to  reduce  the  difficulty  of  mathematical  study 
by  improving  the  seminary  arrangements  and  equipments. 
Not  only  have  special  seminary  libraries  been  formed,  but 
study  rooms  have  been  set  aside  in  which  these  libraries 
are  immediately  accessible  to  the  students.  Collections  of 


AT   THE   GERMAN    UNIVERSITIES. 


109 


mathematical  models  and  courses  in  drawing  are  calculated 
to  disarm,  in  part  at  least,  the  hostility  directed  against  the 
excessive  abstractness  of  the  university  instruction.  And 
while  the  students  find  everywhere  inducements  to  specialized 
study,  as  is  indeed  necessary  if  our  science  is  to  flourish,  yet 
the  tendency  has  at  the  same  time  gained  ground  to  emphasize 
more  and  more  the  mutual  interdependence  of  the  different 
special  branches.  Here  the  individual  can  accomplish  but 
little ;  it  seems  necessary  that  many  co-operate  for  the  same 
purpose.  Such  considerations  have  led  in  recent  years  to  the 
formation  of  a  German  mathematical  association  (Deutsche 
Mathematiker-Vereinigung).  The  first  annual  report  just  issued 
(which  contains  a  detailed  report  on  the  development  of  the 
theory  of  invariants)  and  a  comprehensive  catalogue  of  mathe- 
matical models  and  apparatus  published  at  the  same  time  indi- 
cate the  direction  that  is  here  to  be  followed.  With  the 
present  means  of  publication  and  the  continually  increasing 
number  of  new  memoirs,  it  has  become  almost  impossible  to 
survey  comprehensively  the  different  branches  of  mathematics. 
Hence  it  is  the  object  of  the  association  to  collect,  systema- 
tize, maintain  communication,  in  order  that  the  work  and 
progress  of  the  science  may  not  be  hampered  by  material 
difficulties.  Progress  itself,  however,  remains  —  in  mathe- 
matics even  more  than  in  other  sciences  —  always  the  right 
and  the  achievement  of  the  individual. 

GOTTINGEN,  January,  1893. 


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